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Tecl^nical   Drawing  Series 


Essentials  of  Gear 


ANTHONY 


'1 


Id 


TECHNICAL    DRAWING    SERIES 


THE    ESSENTIALS    OF    GEARING 

A  TEXT  BOOK  FOR  TECHNICAL  STUDENTS  AND  FOR  SELF-INSTRUCTION, 

CONTAINING   NUMEROUS   PROBLEMS  AND 

PRACTICAL  FORMULAS 


GARDNER  C.  ANTHONY,  A.M. 

Prokessor  of  Dravvint.  in  Tufts  College;    Dean  of  the  Bromfield-Pearson  School, 

Author  of  "Elements  of  Mechanical  Drawing,"  and  "  Machine  Drawing;" 

Member  of  American  Society  for  the  Promotion  of  Engineering  Education; 

Member  of  American  Society  of  Mechanical  Engineers 


BOSTON,  U.S.A. 

D.  C.  HEATH  &  CO.,   PUBLISHERS 

1897 


Copyright, 

By  Gardner  C.  Anthony, 

1897. 


PEE  FACE. 


The  most  feasible  method  for  the  acquirement  of  a  working  knowledge  of  the  theory  of 
gear-teeth  curves  is  by  a  graphic  solution  of  problems  relating  thereto.  But  it  requires  much 
time  on  the  part  of  an  instructor,  and  is  very  difficult  for  the  student,  to  devise  suitable  exam- 
ples which,  while  fully  illustrating  the  theory,  shall  involve  the  minimum  amount  of  drawing. 
It  is  tlie  aim  of  the  author  to  overcome  these  difficulties  by  the  presentation  of  a  series  of  pro- 
gressive problems,  designed  to  illustrate  the  principles  set  forth  in  the  text,  and  also  to  encour- 
age a  thorough  investigation  of  the  subject  by  suggesting  lines  of  thought  and  study  beyond 
the  limits  of  this  work. 

In  this  as  in  the  other  books  of  the  series  the  author  would  emphasize  the  fact  that  the 
plates  are  not  intended  for  copies,  but  as  illustrations.  A  definite  lay-out  for  each  problem  is 
given,  and  the  conditions  for  the  same  are  clearly  stated.  This  is  accompanied  by  numerous 
references  to  the  text,  so  that  a  careful  study  of  tlie  subject  is  necessitated  before  performing 
the  problems. 

Although  specially  addressed  to  students  having  no  previous  knowledge  of  the  principles 
of  kinematics,  it  is  also  designed  to  serve  as  supplementary  to  treatises  on  this  subject. 

The  methods  and  problems  have  already  proved  their  usefulness  in  the  instruction  of  stu- 
dents of  many  grades  :   and  it  is  hoped  that  tlieir  publication  may  promote  a  wider  interest  in, 

and  more  thorough  studv  of,  the  essentials  of  gearing. 

GARDNER   C.   ANTHONY. 
Tufts  College,  Sept.  24,  1897. 


2065972 


COISTTEI^TS. 


CHAPTER  I.  PAOK 

Introduction.  —  General  Principles 1 

I.    Constant  Velocity  Ratio.     2.    Positive  Rotation.     3.    (iearing. 

CHAPTER    II. 
Odontoidal  Curves • 4 

4.  Classes  of  Curves.  5.  Cycloid.  6.  Epicycloid.  7.  Hypocycloid.  8.  To  Construct  a  Normal. 
9.  A  Second  Method  for  Describing  the  Cycloidal  Cnrves.  10.  Double  Generation  of  the 
Epicycloid  aud  Hypocycloid.     11.    Epitrochoid.     12.    luvolute. 

CHAPTER    III. 

Spur  Gears  and  the  Cycloidal  System 8 

13.  Theory  of  Cycloidal  Action.  14.  Law  of  Tooth  Coutact.  15.  Application.  16.  Spur  Gears. 
17.  Circular  Pitch.  18.  Diameter  Pitch.  19.  Face  or  Addendum.  20.  Flank  or  De- 
deudum.  21.  Path  of  Contact.  22.  Arc  of  Contact.  23.  Arcs  of  Approach  and  Recess. 
24.  Angle  of  Obliquity  or  Pressure.  25.  Rack.  26.  Spur  Gears  Having  Action  on  Both 
Sides  of  the  Pitch  Point.  27.  Clearance.  28.  Curve  of  Least  Clearance.  29.  Backlash. 
30.  Conditions  Governing  the  Practical  Case.  31.  Proportions  of  Standard  Tooth.  32.  In- 
fluence of  the   Diameter  of   the   Rolling  Circle  on  the   Shai>e   and   Efticiency  of  (iear  Teeth. 

v 


VI  rONTENTS. 

PAGE 

33.  Intercliaii,n'eal)le  (xears.  34.  Practical  Case  of  Cycloidal  Gearing.  35.  Face  of  (iear. 
36.  Comparison  of  Gears,  illustrated  in  Plates  4,  5,  and  6.  37.  Conventional  Representa- 
tion of  Spur  Gears. 

CHAPTER   TV. 

Involute  System 26 

38.  Theory  of  Involute  Action.  39.  Character  of  the  Curve.  40.  Involute  Limiting  Case.  41.  Epi- 
cycloidal  J^xtension  of  Involute  Teeth.  42.  Involute  Practical  Case.  43.  Interference. 
44.  Influence  of  the  Angle  of  Pressure.  45.  Method  for  Determining  the  Least  Angle  of 
Pressure  for  a  Given  Number  of  Teeth  Having  no  Interference.  46.  Defects  of  a  System 
of  Involute  Gearing.     47.    Unsymmetrical  Teeth. 

CHAPTER   V. 

Annular  Gearing 38 

48.  Cycloidal  System  of  Annular  Gearing.  49.  Limiting  Case.  50.  Secondary  Action  in  Annular 
Gearing.  51.  Limitations  of  the  Intermediate  Describing  Curve.  52.  Limitations  of  Exte- 
rior and  Interior  Describing  Curves.  53.  The  Limiting  Values  of  the  Exterior,  Interior,  and 
Intermediate  Describing  Circles  for  Secondary  Action.  54.  Practical  Case.  55.  Summary 
of  Limitations  and  Practical  Considerations.     56.    Involute  System  of  Annular  Gearing. 

CHAPTER    VL 

Bevel  Gearing 45 

57.  Theory  of  Bevel  Crearing.  58.  Cliaracter  of  Curves  Employed  in  Bevel  Gearing.  59.  Tredgold 
Approximation.  60.  Drafting  the  Bevel  Gear.  61.  Figuring  the  Bevel  Gear  with  Axes  at 
90°.     62.    Bevel  (Jeai'  Table  for  Shafts  at  90°.     63-    Bevel  (iears  with  Axes  at  Any  Angle. 


CONTENTS.  VU 


CTIAPTKK    VII. 


I'AGK 


Special  Forms  of  Gears,  Notation,  Formulas,  etc 57 

64.  Odoutograplis  and  Odoiitograph  Tables.  65.  Willis's  Odoiitograph.  66.  The  Tliree-Poiiit 
Odoutograph.  67.  The  (iraiit  Involute  Oduntograph.  68.  The  llobinsuu  Odoutograph. 
69.  The  Klein  C'ooidinate  Odoutograph.  70.  Special  Forms  of  Odontoids.  and  Their  Lines 
of  Action.  71.  Conjugate  Curves.  72.  \\'orni  (Jeaiiug.  "73.  Literature.  74.  Notation 
and  Formulas. 

CHAPTER    YIIL 

Problems ...,.,.,,  ,68 

75.    ^Method  to  l>e  Observed  in  Performing  the  Problems  — 

PuoiiLK.M   1.    Cycloidal  Limiting  Case.     Face  or  Flanlc  Only 69 

'2.    Cycloidal  Limiting  Case.      Face  and  Flank 71 

o.    Cycloidal  (iear.      Practical  Case 72 

4.  Involute  Limiting  Case , .     .  74 

5.  Involute  Practical  Cases 75 

6.  Cycloidal  Annular  (iear 76 

7.  Involute  Annular  Gear 77 

8.  Cycloidal  and  Involute  Bevel  Gears.     Shafts  at  !H)- 78 

9.  Cycloidal  and  Involute  Bevel  (iears.     Shafts  at  Other  than  90" 79 


ERRATA    FOR    THE    ESSENTIALS    OF    GEARING. 

0.5 
Page    l(i,    Hue    10,       For    o.oP    read    -— 

Page  18,  line    10.       For   one-sixth    read    one-seventh. 

Page  18,  line    11.       For    Fig.    6    read    Fig.    9. 

Page  34,  line     4.       For    —    read  ^^ 

Page  69,  problem    1,    column    (i.      For    A    read    a. 

Plate  2.  For   A'    on    director    circle    read    A. 

Plate  10.  Erase    "Pitch    line"    near    H  K. 

Plate  14.  Fig.    '2.     The    pinion    is   designed    to    have    5    teeth    and    the    gear    fi    teeth. 


THE   ESSENTIALS   OF   GEARING. 


CHAPTER   I. 

INTRODUCTION.  —  GENERAL    PRINCIPLES. 

1.  Constant  Velocity  Ratio.  Motion  may  be  transmitted  between  lines  of  shafting  by 
means  of  friction  surfaces  ;  and  if  there  be  no  slipping  of  the  contact  surfaces,  the  circumference 
of  the  one  will  have  the  same  velocity  as  the  circumference  of  the  other.  The  number  of  revo- 
lutions of  the  shafts  will  be  inversely  proportional  to  the  diameter  of  the  friction  surfaces,  and 
this  ratio  will  l)e  maintained  constant  under  the  condition  of  no  slip.  Such  friction  surfaces 
and  shafts  are  said  to  have  a  co7istant  velocity  ratio. 

2.  Positive  Rotation.  In  order  to  transmit  force,  as  well  as  motion,  and  to  insure  its 
being  positive,  it  will  be  necessary  to  place  cogs,  or  elevations,  on  one  of  the  friction  sur- 
faces, and  make  suitable  depressions  in  the  other  surface. 

3.  Gearing.  The  study  of  toothed  gearing  is  a  study  of  the  shape  of  these  cogs,  teeth,  or 
odontoids,  which  are  designed  to  produce  a  positire  rotation  Mhile  preserving  the  condition  of 
constant  velocity  ratio. 

1 


GEARS    CLASSIFIED. 


Fig.  1. 


Fig.  2. 


Fig.  3. 


Fig.  4. 


Gears  may  l)e  classified  as  fol- 
lows :  — 

1.  If  the  shafts  are  parallel, 
the  friction  surfaces  would  be 
cylinders  (Fig.  1),  and  the  gears 
designed  to  produce  the  same 
condition,  as  to  the  velocit}^  are 
called  Spur   G-ears  (Fig.   2). 

2.  If  the  shafts  intersect,  the 
friction  surfaces  would  he  cones 
(Fig.  3),  and  the  gears  called 
Bevel   Gears  (Fig.  4). 

3.  If  the  shafts  are  neither  in- 
tersecting nor  })arallel,  the  friction 
surfaces  will  be  hyperboloids  of 
revolution  (Fig.  5),  and  the  gears 
called  Ilt/perbolic,  or  Skeiv  Grears 
(Fig.  6). 

In  the  preceding  cases  the  ele- 
ments of  the  teeth  are  rectilinear, 
and  the  friction  surfaces  touch 
each  other  along  right  lines. 

4.  If  the  elements  of  the  teeth 


GEARS    CLASSIFIED. 


in  either  of  the  first  three  cases 
l)e  made  helical,  an  entirely  dif- 
ferent class  of  gearing  will  result. 
The  various  forms  are  known  as 
Tivisted,  Sph-(d,  Worm^  and  Screw 
Crearing  (Figs.  7  and  8).  The 
action  of  the  latter  is  analogous  to 
that  of  a  screw  and  nut. 

One  of  these  forms  is  generally 
eni[)lo_yed  as  a  siibstitnte  for  hy- 
perbolic, or  skew  gears,  by  reason 
of  the  ditficnlty  experienced  in  cor- 
rectly forming  the  teeth  of  snch 
gears. 

5.  AnothtM-,  allliough  bnt  little 
used,  form,  is  that  known  as  Fare 
(rf(irut(/.  The  teeth  are  iisnally 
pins  secnred  to  the  face  of  cir- 
cnlar  disks  having  axes  perpen- 
dicnlar.  The  action  takes  place 
at  a  point  only. 

None  of  the  latter  forms  can 
be  represented  by  friction  surfaces. 


Fig.  7. 


Fig.  8. 


4  ODUNTOIDAL    CUKVEtS. 

CHAPTER   II. 

ODONTOIDAL      CURVES. 

4.  The  two  classes  of  ciirves  comnioiily  employed  in  gear  teeth  are  the  cycloidal  and  the 
involute.  A  knowledge  of  their  characteristics  and  methods  of  generating  is  essential  to  an 
understanding  of  their  application  in  gearing. 

5.  Cycloid.  Plate  1,  Fig.  1.  The  cjcloid  is  a  curve  generated  b}-  a  point  in  the 
circumference  of  a  circle  A\hich  rolls  upon  its  tangent.  The  circle  is  called  the  describing, 
or  generating  circle,  and  the  point  is  known  as  the  describing,  or  generating  point.  In 
Fig.  1,  Plate  1,  B  is  the  describing  point,  and  B  D  C  E  the  describing  circle,  which  rolls  on 
its  tangent  E  B'". 

Assume  a  point,  C,  on  the  describing  circle,  and  conceive  the  motion  of  the  circle  to  be 
from  left  to  I'iglit.  As  it  rolls  upon  its  tangent,  the  arc  E  C  will  be  measured  off  on  E  B'" 
until  point  C  becomes  a  point  of  tangency  at  C.  The  center  of  the  describing  circle  will  now 
lie  at  A',  in  the  perpendicular  to  E  B"'  at  C\ 

From  center  A',  with  radius  of  describing  circle,  draw  the  ncAV  position  of  describing  circle. 
The  generating  point  must  lie  in  this  circle  at  a  distance  from  C  equal  to  the  chord  B  C. 

Therefore,  with  radius  equal  to  this  chord,  from  center  C,  describe  an  arc  intersecting  the 
new  position  of  the  describing  circle.  The  line  B'  C  is  called  the  instantaneous  radius,  or 
normaU  of  the  curve  at  B',  it  being  a  [)erpendicular  to  the  tangent  of  the  curve  at  this  point. 


CYCLOIDAL    (UKVES.  5 

The  normal  at  B"  would  be  B"  D'.  The  radius  A'  B'  is  known  as  the  dt'scrilniif/,  or  f/enerating 
radius,  and  A'  C  is  tlie  contact  radius,  or  the  radius  at  the  point  of  eontai-t.  In  like  manner 
other  positions  of  the  describing  pt)int  may  be  found,  and  the  curve  connecting  them  will  l)e 
the  cycloid  required. 

6.  Epicycloid.  Plate  1,  F'ig.  2.  If  the  describing  circle  rolls  upon  the  outside  of  an 
arc,  or  circle,  called  the  director  circle,  the  curve  generated  will  be  an  epicycloid,  Fig.  2, 
Plate  1.  The  method  of  descri1)ing  this  curve  is  similar  to  that  for  the  cycloid,  and  the 
lettering  is  the  same.  It  must  be  observed,  however,  that  any  contact  radius,  as  A'  C,  is  a 
radial  line  of  the  circle  on  which  it  rolls. 

7.  Hypocycloid.  Plate  1,  Fig.  3.  If  the  describing  circle  rolls  upon  tlie  inside  of  a  cir- 
cle, the  curve  generated  will  be  an  hypocycloid.  P'ig.  3  illustrates  this  curve,  the  same  letter- 
ing Ijeing  used  as  that  of  the  preceding  cases. 

If  it  be  re(j[uired  to  draw  a  normal  at  any  point  of  this,  or  the  two  preceding  curves,  the  fol- 
lowing method  may  l)e  employed  :  — 

8.  To  Construct  a  Normal.  From  the  given  point  on  the  curve,  as  a  center,  with  radius  of 
generating  circle,  describe  an  arc  cutting  the  path  described  by  the  center  of  the  generating 
circle.  From  this  point  draw  the  contact  radius,  thus  obtaining  the  contact  point.  Connect 
this  with  the  given  point,  and  tlie  line  will  be  the  required  normal. 

9.  A  Second  Method  for  Describing  the  Cycloidal  Curves.  Plate  2,  Fig.  1.  A  B  C  is  a 
director  circle,  A  D  E,  the  generating  circle  for  the  epicycloid  A  A'  A"  H ,  and  A  K  L  the  generating 
circle  for  the  hypoc3'cloid  A  L  C. 


b  CYCLOIDAL    (  UKVES. 

To  describe  the  epicycloid,  assume  any  point,  D,  on  tlie  generating  circle,  and  lay  off  the  arc 
A  D'  on  the  director  circle,  making  it  equal  to  arc  A  D.  If  A  be  the  describing  point,  then  A  D 
will  be  the  normal  when  D  shall  have  become  a  contact  point,  as  at  D'.  With  L  as  a  center,  de- 
scribe the  arc  D  A',  llie  describing  point  A  must  be  in  this  arc  when  D  shall  be  at  D'.  From  D' 
as  a  center,  with  radius  equal  to  the  chord  A  D ,  describe  an  arc  intersecting  A'  D ,  and  thus  deter- 
mine A',  a  point  in  the  epicycloid.     Similarly  obtain  other  points,  and  draw  the  curve. 

The  hypocycloid  may  be  constructed  in  like  manner,  as  shown  by  the  same  figure.  This 
also  illustrates  a  special  case  in  which  the  hypocycloid  is  a  radial  line,  A  L  C,  and  this  is  due 
to  the  diameter  of  tlie  describing  circle  being  equal  to  the  radius  of  the  director  circle. 

The  same  method  may  also  be  employed  in  the  construction  of  the  cycloid. 

10.  Double  Generation  of  the  Epicycloid  and  Hypocycloid.  Plate  2,  Fig.  1.  The  epi- 
cycloid may  always  be  generated  by  either  of  two  describing  circles,  which  differ  in  diameter 
by  an  amount  equal  to  the  diameter  of  the  director  circle.  Thus  in  the  case  illustrated,  the 
epicycloid  A  A'  A"  H  may  be  generated  by  the  circle  A  D  E ,  with  A  as  a  describing  point,  or  by 
the  circle  s  T  H ,  with  H  as  a  describing  point.  Similarly  the  hypocycloid  is  capable  of  being- 
generated  by  either  of  two  rolling  circles,  the  sum  of  which  diameters  must  equal  that  of  the 
director  circle.* 

11.  Epitrochoid.  Plate  2,  Fig.  2,  When  the  describing  point  does  not  lie  on  the  cir- 
cumference of  the  generating  circle,  a  curve,  connnonly  called  an  epitrochoid,  is  described.  If 
the  point  lies  without  the  circle,  as  at  B ,  a  looped  curve,  B  B'  B",  called  the  curtate  epitrochoid, 

*  For  tlie  geoiiietriiMl  (Iriuuij.sUatiuu  of  this  ])r'>l)lein  see  tUe  aiipeiulix  of  Professor  jNIacCord's  ■"  Kinematics,"  page  old. 


INVOLUTE    CURVE.  7 

is  described:  and  if   the  point  be  within,  as  at  D,  the  curve  will  be  a  prolate  epitrot-hoid,  as 
D  D'  D". 

To  obtain  a  point  in  the  former,  assume  any  point,  C,  in  the  circumference  of  the  describing 
circle,  and  determine  its  position,  C,  when  it  shall  have  become  a  contact  point.  Draw  the 
contact  radius  A'  C,  and  from  C  and  A'  as  centei-s,  with  radii  A  B  and  C  B,  describe  arcs  intersect- 
ing at  B',  a  point  in  the  curve.  B'  C  is  the  normal  at  this  point.  In  like  manner  obtain  the 
point  D'  in  the  })rolate  epi trochoid. 

12.  Involute.  Plate  1,  Fig.  4.  The  involute  is  a  curve  generated  by  a  point  of  a  tan- 
gent right  line  rolling  upon  a  circle,  known  as  the  base  circle,  or  the  describing  point  may  be 
regarded  as  at  the  extremity  of  a  fine  wire  which  is  unwound  from  a  cylinder  corresponding  to 
the  base  circle.  In  Fig.  4,  A  B  C  D  is  the  arc  of  a  base  circle,  and  A  the  point  from  which  the 
involute  is  generated. 

Layoff  arcs  A  B,  B  C,  C  D,  preferabh'  equal  to  each  other,  and  from  points  B,  C.  and  D,  draw 
tangents  equal  in  length  to  the  arcs  A  B,  A  C,  and  A  D.  A  line  drawn  through  the  extremity  of 
these  tangents  will  be  an  involute  of  the  base  cii-cle  A  B  C  D. 


b  THE    CYCLOIDAL    SYSTEM. 

CHAPTER   III. 

SPUR  GEARS  AND  THE  CYCLOIDAL  SYSTEM. 

13.  Theory  of  Cycloidal  Action.  Plate  3,  Fig.  1.  ].iet  H  K  and  M  L  be  the  peripheries 
of  two  disks,  having  centers  G  and  F,  and  S  the  center  of  a  third  disk,  also  revolving  in  contact 
with  the  arcs  H  K  and  M  L.  The  largest  disk  will  be  known  as  disk  1,  the  second  size  as  disk 
2,  and  the  smallest  as  disk  3,  or  the  describing  disk.  Consider  the  ^peripheries  of  these  disks  in 
contact  at  A,  so  that  motion  imparted  to  one  will  produce  an  equal  motion  in  the  circumference 
of  the  other  two,  thus  maintaining  at  all  times  an  equal  circumferential  velocity,  or  constant 
velocity  ratio. 

Imagine  this  to  represent  a  model,  disk  1  having  a  flange  I  0  extending  below  the  other 
disks,  and  the  describing  disk  as  being  provided  with  a  marking  point  at  A,  each  of  the  disks 
being  free  to  revolve  about  their  respective  axes.  Consider  first  the  relation  between  the 
describing  disk  and  disk  1,  the  marking  point  being  at  A.  Suppose  motion  to  be  given  disk  3 
in  the  direction  indicated  by  the  arrow,  so  that  the  describing  point  shall  move  from  A  to  A'. 
The  point  C  of  disk  1,  which  coincides  with  A  when  the  describing  disk  is  in  the  first  position, 
will  now  have  moved  on  the  circumference  H  K,  to  C,  an  arc  equal  to  A  A'.  During  this  time, 
the  curve  A'  C  will  have  been  drawn  upon  the  flange  of  disk  1  by  the  marking  point.  Next, 
suppose  the  marking  point  to  move  from  A'  to  A",  then,  since  the  circumferences  of  these  disks 
tra\erse  equal  spaces  in  equal  times,  C  will  have  revolved  to  C",  and  the  curve  A'  C  will  now 
(ic'fupy  the  position  E"  C".     But,  since  the  niarking  point  has  continued  to  describe  a,  curve  upon 


CYCLOIDAL    ACTION.  9 

the  flange  oi  cli.sic  1.  the  curve  E"  C"  will  be  extended  to  A".  In  like  manner  the  marking-  point 
moves  to  A"',  continuing  to  describe  a  curve,  as  C"  A"  revolves  to  C"  A'".  If  now  the  describing 
disk  be  freed  from  the  axis  on  which  we  have  supposed  it  to  revolve,  and  Ije  rolled  on  the  cir- 
cumference H  K,  the  marking  point  would  describe  the  same  curve,  h'"  E'"  C",  as  that  already 
drawn,  which  is  an  epicycloid. 

In  the  same  manner,  we  may  imagine  the  marking  point  to  describe  a  curve  upon  disk  2, 
which  curve,  in  its  successive  positions,  would  be  shown  by  A'  B',  A"  B",  and  A'"  B'".  For  the 
same  reason,  too,  the  arc  A  A'  A"  A'"  will  equal  the  arc  B  B'  B"  B"' ;  and  if,  in  a  manner  similar  to 
the  preceding,  we  roll  the  describing  disk  on  the  inside  of  the  arc  M  L,  we  shall  describe  the 
same  curve  A'"  D'"  B'",  and  find  it  to  be  an  hypocycloid. 

Again,  consider  these  curves,  A'"  C"  and  A'"  B'",  as  l)eing  traced  at  the  same  time  l)y  the 
describing  point  A.  If  we  now  observe  any  special  position  of  the  point,  as  A",  it  will  be  seen 
to  be  connnon  to  an  epicycloidal,  and  a  hypocycloidal  curve,  which  have  a  common  normal, 
A"  A,  intersecting  the  line  drawn  through  the  centers,  F  and  G,  at  the  point  of  tangency  of  the 
disks.     This  condition  is  true  for  all  positions  of  the  two  curves. 

If  these  curves  A'"  C"\  and  A'"  B'",  be  now  used  as  the  outlines  for  gear  teeth,  as  in  Fig.  2,  G 
and  F  being  the  centers  and  H  K  and  M  L  the  pitch  lines,  we  shall  have  obtained  a  positive  rota- 
tion with  a  uniform  velocity  ratio,  for  it  was  under  this  latter  condition  that  the  curves  were 
generated,  and  the  connnon  normal  to  the  curves  at  any  point  of  contact  will  pass  through  the 
point  A  (the  pitch  point).     Such  curves  are  said  to  be  conjugate. 

It  is  not  necessary  that  the  describing  point  be  on  the  circumference  of  the  circle,  or  that 
the  describing  curve  be  a  circle,  in  order  to  obtain  two  curves  which,  acting  together,  shall  pro- 
duce a  constant  velocity  ratio. 


10  LAW    OF    TOOTH    CONTACT    AM)    .SPl'K    (JEAHS. 

14.  Law  of  Tooth  Contact.  In  order  to  preserve  the  eoiiditiou  of  coiistaiit  velocity  ratio, 
the  tootli  outlines  which  act  in  contact  must  he  such  that  the  common  normals  at  the  point  of 
contact  shall  always  cut  the  line  of  centers  in  the  same  point ;  and  in  general,  tiie  curves  must 
1)6  such  as  may  be  simultaneously  traced  upon  the  planes  of  rotation  of  two  disks,  while  re- 
volving, by  a  marking  point  which  is  carried  by  a  describing  curve,  moving  in  rolling  contact 
with  both  disks. 

15.  Application.  Suppose  action  to  take  place  between  the  odontoids,  or  gear  teotli,  shovvii 
in  Fig.  2,  Plate  3.  Let  1  l)e  the  driver,  and  suppose  motion  to  begin  from  the  position  shown 
in  the  figure,  the  contact  being  at  A.  As  the  motion  takes  place,  points  A',  A",  A'",  will  succes- 
sively come  into  contact,  their  common  normals  passing  through  the  PITCH  POINT,  A,  at  the 
time  of  their  contact,  thus  producing  a  constant  velocity  ratio,  and  the  periphery,  or  pitch 
LINE,  of  1,  will  have  the  same  velocity  as  the  periphery,  or  pitch  line,  of  2.  But  this  uniform 
motion  must  cease  when  points  k'"  W"  come  into  contact,  and  the  velocity  ratio  will  remain  con- 
stant no  longer,  unless  a  second  pair  of  curves  begin  contact  at  this  moment. 

Plate  4  illustrates  a  pair  of  disks  provided  with  a  series  of  these  curves  arranged  so  as  to 
continue  the  motion  indefinitely  in  eitlier  direction. 

16.  Spur  Gears.  Plate  4.  F  is  the  center  of  a  pinion  having  twelve  teeth,  and  G  the 
center  of  a  gear  of  eighteen  teeth,  only  a  segment  of  the  latter  being  shown.  A  C  K  is  the 
describing  circle,  carrying  the  marking  point  C,  which  descril)ed  the  epicycloid  C  D,  and 
the  hypocycloid  C  E.  The  depth  of  the  pinion  tooth  must  be  made  sufficient  to  admit  the 
addendum  of  the  gear  tooth,  but  only  that  portion  of  the  curve  between  C  and  E  will  engage 


CIRCULAR    AND    DIAMETER    PITCH.  11 

C  D.  The  reinaiiider  of  the  pinion  Hank  may  l)e  a  continuation  of  tlio  hypocN^chiicI,  or  any 
other  curve  which  may  not  interfere  with  the  action  of  the  gear  tooth.  The  oppoSi.te  sides  of 
the  teeth  are  made  alike  in  order  that  motion  may  take  place  in  either  direction.  If  the  direc- 
tion be  that  indicated  by  the  arrows,  the  pinion  bemg  the  driver,  the  sluuled  side  of  the  teeth 
would  have  contact ;   and  if  the  direction  be  reversed  the  opposite  faces  would  engage. 

In  order  to  accurately  reproduce  the  dedenda  of  the  pinion,  a  scroll  may  be  used  in  the  fol- 
lowing manner :  — 

Having  selected  one  to  match  the  tooth  curve,  C  E,  continue  the  curve  of  the  scroll  by  the 
center  F,  from  whicli  a  circle  should  l)e  drawn  tangent  to  the  line  of  the  scroll.  jNlark  that 
point  of  the  scroll  in  contact  with  the  pitch  circle.  Having  laid  off  the  pitch,  and  thickness  of 
the  teeth,  place  the  marked  point  of  the  scroll  to  coincide  with  these  points,  and  at  the  same 
time  tangent  to  the  circle  already  drawn.  Draw  such  part  of  the  curve  as  lies  between  the 
addendum  and  dedendum  circles.  Reverse  the  scroll  for  drawing  the  opposite  side  of  the 
teeth. 

17.  Circular  Pitch.  The  distance  A  D,  or  A  E,  measured  on  the  pitch  line  between  cor- 
responding  points    of    consecutive    teeth,    is    called    the    circular    pitch,    and    is    equal   to  the 

circumference  ot  pitch  circle 
iiuiiiber  ol  teeth 

Let  P'  denote  the  circular  pitch,  D'  the  diameter  of  the  pitch  circle,  and  N  the  numl)er  of 
teeth,  then  will   P'  =  '^  (1),  and,  p- =  ^  (2). 

18.  Diameter  Pitch.  In  order  to  ex})ress  in  a  more  direct  and  simj)le  manner  the  ratio 
between  the  diameter  of  the  pitch  circle  and  the  number  of  teeth,  and  to  easily  determine  the 


12  TOOTH    PARTS. 

proportions  of  the  teeth,  it  hiis  been  found  expedient  to  a})ply  the  term  pitch,  or  more  properly, 
diameter  pitch,  to  designate  the  ratio  between  the  number  of  teeth  and  tlie  diameter  of  pitch 
circle.  This  is  not  an  ahsolufe  mrasvre^  hut  a  ratio  ;  and  since  it  may  usually  be  expressed  by  a 
whole  number,  the  proportions  of  the  parts  of  a  tooth,  which  are  commonly  dependent  on 
the  pitch,  may  be  more  readily  determined,  and  all  the  figuring  of  the  gear  simplified. 

Designating  the  diameter  pitch  l)y  P,  P  =  ^,  (3). 

To  obtain  the  relation  between  the  diameter  pitch  and  the  circular  pitch,  compare  formulas 
2  and  3.  -=— ,  —  =  ?;  lience^,  =  P  or  P  P'  =  7r(4).  'J'liis  last  e(piation  expresses  the  I'elation 
between  tlie  two  pitches  in  a  simple  form  which  may  ])e  easily  remembered. 

Illustration.  —  Tlie  pinion  represented  in  Plate  4  has  12  teeth,  and  is  3  inches  in 
diameter,  k;  =  P?  T  ""  ^-  '^'^^^  pitch,  therefore,  is  4.  The  circular  pitch,  P'  =  □  =  —7 —  =  .7854. 
Having  given  any  two  of  the  terms,  N,  D',  P,  P',  the  other  terms  may  l)e  determined. 

19.  Face,  or  Addendum.  Tliat  portion  of  the  tooth  curve  lying  outside  of  the  pitch  circle 
is  called  the  face  or  adtlendum,  as  C  D,  Plate  4. 

20.  Flank,  or  Dedendum.  That  portion  of  the  tooth  curve  lying  inside  of  tht^  pitch  circle 
is  called  the  flaidc  or  dedendum,  as  E  H ,  Platp:  4. 

21.  Path  of  Contact.  In  Fig.  1,  Plate  3,  it  will  be  observed  that  the  contact  between 
the  two  curves  takes  place  in  the  arc  A  A'  k"  A'".  This  is  called  the  path  of  contact,  or  line 
of  action,  and  in  the  C3-cloidal  system  this  line  is  an  arc  of  the  describing  circle. 


ARCS    OF    CONTArT.  13 

22.  Arc  of  Contact.  The  tire  (leseril)e(l  l)y  a  point  on  the  pitch  lin(;  during'  the  time  of  con- 
tact of  two  odontoids  is  called  the  arc  of  contact.  It  must  not  be  less  than  the  pitch.  In  this 
case  the  arc  of  contact  would  be  measured  by  the  arcs  A  D  or  A  E,  and  these  arcs  being  equal  to 
the  pitch,  the  case  is  called  a  limiting  one.  In  ^)ractice  it  should  be  greater,  wliicli  w^ould  be 
accomplished  by  lengthening  the  addendum. 

23.  Arcs  of  Approach  and  Recess.  There  are  four  cases  of  contact  that  may  take  place 
between  the  gear  and  pinion  of  Plate  4. 

1.  Gear  as  driver.     Direction  opposed  to  the  arrows.     Contact  begins  at  A  and  ends  at  C. 

2.  Pinion  as  driver.      Direction  same  as  arrows.     Contact  begins  at  C  and  ends  at  A. 

In  each  of  these  cases  the  action  will  take  place  between  the  shaded  portions  of  the  teeth. 

3.  Gear  as  driver.     Direction  same  as  arrows.     Contact  begins  at  A  and  ends  at  L. 

4.  Pinion  as  driver.     Direction  opposed  to  the  arrows.     Contact  begins  at  L  and  ends  at  A. 
In  the  last  two  cases  there  will  be  no  contact  between  the  shaded  portions  of  the  teeth. 

In  the  lii'st  and  third  cases  the  contact  takes  place  from  the  pitch  point,  and  is  called  an  arc 
of  recess. 

In  the  second  and  fourth  cases  the  contact  takes  place  tow^ard  the  pitch  point,  ending  at  A, 
and  is  called  an  arc  of  approach. 

It  should  also  be  observed  that  in  the  case  illustrated  the  arc  of  contact  must  be  either  one 
of  approach  or  of  recess;  but  had  the  teeth  of  each  gear  been  provided  with  curves  on  both 
sides  of  the  pitch-line,  as  in  Plate  5,  the  arc  of  contact  would  have  consisted  of  an  arc  of 
approach  and  of  recess.  (See  Art.  30,  page  10,  for  a  further  discussion  of  the  relation  between 
these  arcs.) 


14  ANGLE    OF    PRIiSSURE. RACK. 

24.  Angle  of  Obliquity,  or  Pressure.  Tlie  angle  which  the  conimoii  normal  to  a  pair  of 
conjugate  teeth  makes  with  the  tangent  at  the  pitch  point,  is  called  the  angle  of  obliquity,  or 
angle  of  pressure.  The  angle  CAP,  Plate  4,  is  the  angle  of  greatest  obliquity.  The  greater 
this  angle,  the  greater  the  tendency  to  thrust  the  gears  apart ;  the  friction  will  be  increased 
and  the  component  of  force  tending  to  produce  rotation  will  be  decreased. 

25.  Rack.  If  the  diameter  of  the  gear  be  indefinitely  increased,  the  pitch  circle  will  finally 
become  a  right  line,  and  the  gear  will  then  be  known  as  a  rack. 

The  rack  shown  in  Plate  4  has  teeth  only  on  one  side  of  the  pitch  line,  like  the  pinion  and 
gear,  and  the  conditions  of  action  are  similar.  The  tooth-curve  will  be  a  cycloid,  and  the 
rolling  circle,  M  N  0,  must  be  the  same  as  that  used  for  the  engaging  pinion,  in  order  to  fulfil 
the  general  law  for  maintaining  a  constant  velocity  ratio  (Art.  14,  page  10). 

26.  Spur  Gears  having  action  on  both  sides  of  the  Pitch  Point,  Plate  5.  If  we  assume  the 
diameters  of  pitch  and  rolling  circles  to  be  the  same  as  before,  and  the  arc  of  action,  C  A,  un- 
changed, tlie  addendum  of  gear  and  dedendum  of  pinion  will  be  the  same  as  those  of  Plate  4. 
This  case,  however,  differs  from  the  preceding  in  that  the  number  of  teeth  is  but  half  as  great, 
and  therefore  the  pitch  will  be  doubled.  This  will  require  the  arc  of  action  to  be  doul)led,  in 
order  that  it  shall  equal  the  pitch  (Art.  22,  page  13).  Such  increase  in  the  arc  of  action  may 
be  made  by  continuing  the  path  of  contact  to  the  other  side  of  the  pitch  point,  following  the 
circumference  of  a  rolling  circle  which  may  or  may  not  be  equal  to  the  other  rolling  circle. 
Having  laid  off  the  arc  A  H  equal  to  one-half  the  circular  pitch,  describe  the  curves  H  K  and 
H  L ,  with  H  as  the  generating  point  of  the  new  rolling  circle.     The  former  of  these  curves  will 


(UllVE    OF    LEAST    CLEAIIAXCE.  15 

beL-uiiiL'  the  addiMuliini  of  the  pinion,  ami  the  Litter  the  dedencliini  of  the  gear  tooth.  The  en- 
gaging gears  ^vill  then  liave  both  faces  and  flanks,  the  action  ^vill  l)egin  at  C  and  end  at  H ,  the 
path  of  contact  will  be  C  A  H,  the  arc  C  A  being  the  path  of  approach,  and  A  H  the  path  of 
recess,  their  snm  being  equal  to  the  circular  pitch. 

In  a  similar  manner  the  dedendum  of  the  rack  tooth  may  be  described  to  engage  the  adden- 
dum of  the  pinion  tooth,  and  the  contact  begun  at  N  Avill  end  at  0,  N  M  being  the  path  of 
approach,  and  M  0  the  path  of  recess.  That  portion  of  the  dedendum  of  rack  tootli  which 
engages  the  addendum  of  the  pinion  is  indicated  b}-  sectioning,  but  it  is  necessary  to  continue 
the  dedendum  to  a  depth  sufficient  to  allow  the  addendum  of  the  engaging  tooth  to  enter. 

27.  Clearance.  The  space  between  the  addendum  circle  of  one  gear  and  the  dedendum 
circle  of  an  engaging  gear  is  called  clearance.     Fig.  9,  page  17. 

28.  Curve  of  Least  Clearance.  If  the  pitch  circle  of  the  gear  ])e  rolled  on  that  of  the 
pinion,  and  the  epitrochoid  of  the  highest  point,  C,  of  the  gear  tooth  be  determined,  it  will  be 
the  curve  of  least  clearance. 

The  successive  positions  of  the  tootli,  when  so  revolved,  are  sliown  by  the  dotted  line  in 
Plate  5,  and  the  line  connecting  these  points  wotdd  ])e  the  desired  curve.  This  nui}^  be 
obtained  as  follows :  Assume  any  point,  R  on  the  pitch  circle  of  pinion,  and  la^•  off  ai'c  A  R'  on 
the  pitch  circle  of  gear,  equal  to  arc  A  R.  From  R,  with  radius  R  C,  equal  to  R'  C,  describe  an 
arc.  Similarly  describe  other  arcs,  and  draw  a  curve  touching  these  arcs  on  the  inside.  This 
curve  will  be  tlie  curve  of  least  clearance.* 

*  See  also  the  method  of  Akt.  71,  page  til. 


16  CONDITIONS    (;()A^Ei:\lN(;    TIIK    PRACTICAL    CASE. 

29.  Backlash.  In  order  to  allow  for  unavoidable  inaccuracies  of  workmanship  and  operat- 
ing, it  is  customary  to  make  the  sum  of  the  thickness  of  two  conjugate  teeth  something  less 
than  the  circular  pitch.      This  insures  contact  between  the  engaging  faces  only. 

30.  Conditions  governing  the  practical  case.  From  a  consideration  of  the  foregoing  limiting 
cases,  the  following  principles  are  deduced,  to  which  are  also  added  the  limitations  and  modifi- 
cations established  by  practice. 

1.  The  curves  of  gear  teeth,  which  act  to  produce  a  constant  velocity  ratio,  must  be 
described  by  the  same  circle  rolling  in  contact  with  their  respective  pitch  circles.  (Akt.  14, 
page  10.)  Practical  considerations  limit  the  diameter  of  the  describing  circle  to  a  maximum 
of  about  — ,  or  equal  to  the  radius  of  the  pitch  circle,  and  a  minimum  of  about  1^  P',  or  6.5  P. 

See  also  Art.  32,  page  19. 

2.  The  arc  of  contact  must  equal  the  circular  pitch,  and  in  practice  exceed  it  as  much  as 
possible. 

8.  The  addendum  of  a  gear  tooth  engages  the  dedendum  of  the  pinion,  and  the  action 
between  them  either  begins  or  ceases  at  the  pitch  point. 

Since  the  addendum  and  dedendum  of  any  tooth  are  independent  curves,  they  may  be 
described  by  rolling  circles  differing  in  diameter. 

4.  In  the  limiting  cases  considered,  the  height  of  the  tooth  is  dependent  on  the  arc  of  con- 
tact, but  in  practice,  the  arc  of  contact  is  made  dependent  on  the  height  of  the  tooth. 

While  it  is  an  almost  universal  custom  to  make  the  addenda  of  engaging  teeth  equal,  there 
are  special  cases,  in  which  very  smooth-running  gears  are  required,  where  it  would  be  advan- 
tageous to  make  the  addenda  of  the  driver  less  than  those  of  the  driven  gear,  thus  increasing 
the  arc  of  recess,  or  decreasing  the  arc  of  approach. 


PROPORTIONS  OF  STANDARD  TOOTH. 


17 


The  approacliing  uetion  Ijeing-  more  (letrinientiil. 
by  reason  of  the  friction  induced,  it  is  common  to  de- 
sign clock  gears  so  as  to  eliminate  this  by  providing 
the  driver  with  faces  only,  and  the  driven  with  flanks 
only.  Or,  if  the  gears  are  made  with  both  faces  and 
flanks,  to  so  round  the  faces  of  the  driven  gear  tluit  no 
action  may  take  place. 

31.  Proportions  of  Standard  Tooth.  The  propor- 
tions most  connnonly  accepted  for  cut  gears  are  those 
illustrated  in  Fig.  !'.  The  dimensions  are  made  depen- 
dent on  the  pitch,  as  follows  :  — 


Addendum,  (S)  =  - ■. — 

diam. -pitch 
Dedendum,    (s  +  f)  = 


~r. : — :   +  clearance  =  -  +  f . 

diam   pitch  r 

Thickness,   (t)  =  ^  circular  pitch  =  y  =  7^  • 
Clearance,  (f)  =  ^ addendum  =  |=  ^  or,  f  =  ^^  thick- 

t  P'  TT 

ness  of  tooth  =  —=—:  =  „„   „  • 
iO        20        20   P 

In    assuming   this    value    for   the   thickness    of   the 
tooth   the   backlash  is  taken  as  zero,  but  of  course  the 


Fig.  9. 


18  INFLUENCE    OF    THE    ROLLING    CIRCLE. 

tooth  must  be  slightly  smaller  than  the  space  to  permit  of  freedom  in  action.  If  there  he 
any  backlash  the  value  of  t  will  be  e.rcuiar  pitch -backlash^  j,^  rough  cast  gears  the  backlash  may 
be  as  great  as  ^Vth  the  circular  pitch,  but  this  amount  is  very  excessive.  It  is,  however,  in- 
consistent to  base  the  values  for  backlash,  or  clearance,  on  the  pitch,  since  an  increase  in  the 
size  of  the  tooth,  or  pitch,  does  not  necessarily  mean  a  proportional  increase  in  tlie  allowance 
to  be  made  for  the  inaccuracies  of  workmanship.  Indeed,  l)()th  these  clearances  nuist  be  left  to 
the  judgment  of  the  designer. 

Fillets.  The  circular  arc  tangent  to  the  flank  and  dedendum  circle  is  called  the  fillet.  It 
is  designed  to  strengthen  the  tooth  by  avoiding  the  sharp  corner  at  the  root  of  the  tooth.  A 
good  rule  is  that  of  making  the  radius  of  fillet  equal  to  one-sixth  of  the  space  between  the 
teeth,  measured  on  the  addendum  circle,  as  in  Fig.  6.  Tlie  limit  of  size  may  be  determined  by 
obtaining  the  curve  of  last  clearance.     Art.  28,  page  15. 

32.  Influence  of  the  Diameter  of  the  Rolling  Circle  on  the  Shape  and  Efficiency  of  Gear 
Teeth.  If  the  height  of  the  teeth  be  previously  determined,  any  increase  in  the  diameter  of  the 
describing  circle  will  increase  the  path  of  contact  and  decrease  the  angle  of  pressure.  Bat 
since  an  increase  in  the  diameter  of  the  describing  circle  produces  a  weaker  tooth,  by  reason  of 
the  undercutting  of  the  flank,  as  shown  in  Fig.  12,  page  21,  the  maximum  limit  of  the  diameter 
is  connnonly  made  equal  to  the  radius  of  the  pitch  circle  within  which  it  rolls.  As  was  shown 
iji  Art.  9,  page  5,  this  will  generate  a  radial  flank.  In  the  case  of  gears  designed  to  trans- 
mit a  uniform  force,  and  not  subjected  to  sudden  shocks,  it  is  desirable  that  the  teeth  have 
radial  flanks,  and  consequently  the  diameters  of  the  rolling  circles  will  be  equal  to  the  radii 
of  the  pitch  circles  within  which  they  roll.     If  the  force  to  be  transmitted  be  irregulai-,  and  the 


INFLUENCE    OF    THE    KULLING    (IKCLF. 


10 


teeth  required  to  sustain  sudden  strains,  it  is  better  that 
the  flank  be  made  wider  at  the  dedendum  circle,  and  a 
describing-  circle  chosen  of  a  diameter  sufficiently  small 
to  produce  the  desired  result. 

In  general,  the  diameters  of  the  descril)ing  circles 

D'  5 

will  lie  between  the  values  of  -^  and  - .      The  second 

i.  r 

value  was  used  for  the  describing  circle  of  the  gears  in 
Plate  5,  and  would  describe  radial  flanks  for  a  gear 
having  ten  teeth. 

Fig.  10  illustrates  the  effect  of  a  change  in  the  di- 
ameter of  the  rolling  circle  on  the  path  of  contact  and 
angle  of  pressure.  Two  gears  of  equal  diameter  are 
supposed  to  engage,  and  the  teeth  are  described  by  roll- 
ing circles  of  equal  diameter. 

K  P  is  the  addendum,  and  P  L  the  dedendum  of  the 
tooth  described  b}-  the  rolling  circles  C  P,  and  P  D, 
which  are  of  the  same  diameter,  and  equal  to  one- 
quarter  of  the  pitch  diameter.  A  C  being  the  ad- 
dendum line  of  the  engaging  gear,  C  may  be  considered 
as  the  first,  and  D  as  the  last,  point  of  contact.  The 
arcs  C  P  and  P  D  constitute  the  i)a-th  of  contact,  and  the 
angle  C  P  H  is  the  angle  of  pressure. 

Next  consider  the  describing  circle  as  increased,  its 


Fig.  10. 


20 


INTERCHANGEABLE    GEARS. 


Fig.  11. 


diuineter  being  equal  to  one-half  of  the  diameter  of  the  pitch 
circle.  The  form  of  the  tooth  will  now  be  E  P  F,  and  the  path 
of  contact  A  P  B.  In  the  latter  case  the  arc  of  contact  will 
be  greater,  the  maximum  angle  of  pressure  less,  and  the 
tooth  weaker  than  in  the  former. 

The  relation  between  the  two  cases  may  be  more  exactly 
stated  as  follows  :  — 

Diameter  of  describing  curve,        —  — . 

Arc  of  contact,  i.08  P'     1.35  P'. 

Maximum  angle  of  pressure,       32.3°       20  3\ 

Again,  the  weakness  of  the  tooth  in  the  second  case  may 
be  partially  overcome  by  reducing  the  height  of  the  tooth, 
and  in  general  this  would  be  advantageous,  the  so-called 
standard  tooth  being  too  high  for  the  best  results. 

33.  Interchangeable  Gears.  Since  the  same  diameter  of 
rolling  circle  must  be  used  for  the  addendum  of  pinion  tooth, 
and  the  dedendum  of  engaging  gear  tooth,  it  follows  that  for 
any  system  of  interchangeable  gears,  the  addenda  and  de- 
denda  of  all  teeth  nuist  be  described  by  the  same  descril)ing 
cur\^e.  It  is  also  necessary  that  the  pitch,  and  proportion 
of  the  teeth,  be  constant. 


PRACTICAL    CASE. 


21 


In  pi-aetice,  it  is  common  to  regard  g'eai*s  of  twelve  or  t»f 
fifteen  teeth  as  the  hase  of  the  system,  and  the  diameter  of 
the  rolling  circle  is  made  equal  to  the  radius  of  the  corre- 
sponding pitch  circle,  thus  describing  teeth  with  radial  flanks 
for  the  smallest  gear  of  the  set.  If  twelve  be  adopted  as  the 
smallest  number  of  teeth  in  the  system,  the  diameter  of  the 
pitch  circle  will  be    D'  =  -  =  — ,  and  the  diameter  of  the  de- 

scribino-  circle  will  be  t-t;  =  -• 

&  2  P        P 

Again,  if  a   fifteen-toothed  gear  be   used  as  the  base  of 

7  5 

the  system,  the  diameter  of  the  describing  circle  will  be  — . 

Figs.  11  and  12  illustrate  a  fifteen  and  a  nine-toothed 
gear  engaging  a  rack.  The  diameter  of  the  rolling  circle  by 
which  the  teeth  were  described  is  — ,  which  will  equal  3.75 
inches  for  a  2  pitch  gear. 

The  fifteen-toothed  gear  Mill  have  radial  flanks,  but  the 
nine-toothed  gear  will  have  the  flanks  much  undercut  by 
reason  of  the  diameter  of  the  rolling  circle  exceeding  the  ra- 
dius of  the  pitt-h  circle. 

34.  Practical  Case  of  Cycloidal  Gearing.  Plate  6.  Let 
F  and  G  be  the  centers  of  pinion  and  gear  having  twelve  and 
eighteen  teeth  respectively,  and  a  diameter  pitch  of  4.     The 


Fig.  12. 


22  SPUR    C4EARS. 

pitch  diameters  will  equal  5=7"='"  iiif'lit's,  and  p  ^t  =-l.V  inches  (Art.  18,  page  12).     If  the 

tooth  be  of  standard  dimensions,  the  addendum  and  dedendum  lines  may  be  determined  and 
drawn  by  Ai;t.  31,  page  IT.  The  diameter  of  the  rolling  circle  is  assumed  to  be  1^  inches  for 
the  addendum  and  dedendum  of  both  gears.  Since  the  teeth  should  usually  l^e  shown  in  con- 
tact at  the  })itch  point,  suppose  the  generating  point  of  the  describing  curve  to  be  at  this  point, 
and  describe  the  curves  by  rolling  the  circles  from  this  position,  first  on  the  inside  of  one  pitcli 
circle,  and  then  on  the  outside  of  the  other  pitch  circle,  thus  obtaining  the  flank  of  one  tooth, 
and  the  engaging  face  of  a  tooth  of  the  other  gear. 

An  enlarged  representation  of  these  curves  is  shown  in  Plate  (J.  They  may  be  di-awn  by 
the  methods  of  Arts.  6  and  7,  l^age  5,  or  by  Art.  9,  page  5,  but  care  should  be  used  to  draw 
them  in  their  proper  relation  to  each  other,  as  sliown  in  the  figure,  so  that  it  may  not  be  neces- 
sary to  reverse  the  curves  in  order  to  incor})orate  them  into  tooth  forms.  The  order  for  the 
drawing  of  the  curves  ma}*  be  A  B,  A  T,  A  D,  A  S. 

Instead  of  reproducing  the  tooth  curves  by  means  of  scrolls,  it  is  sufficiently  accurate,  and 
much  more  rapid,  to  approximate  them  by  circular  arcs.  Plate  2,  Fig.  3,  illustrates  a  simple 
method  which  closely  approximates  the  curves  of  this  system,  and  suffices  for  the  ordinary 
drawing  of  a  geai',  but  in  no  case  should  be  used  for  descril)ing  the  curves  for  a  templet.  This 
method  consists,  Jirsf,  in  the  construction  of  a  normal  for  a  point  of  the  curve  at  a  radial 
distance  from  the  pitch  line  equal  to  two-thirds  of  the  addendum  or  dedendum  of  the  tooth; 
8eco)id,  in  the  finding  of  a  center  on  this  normal,  such  that  an  arc  may  Ije  described  through 
the  pitch  point,  and  the  point  of  the  tooth  already  determined.  A  P  is  the  height  of  the  adden- 
dum,  and  B  a  point  radially  distant  from  the  pitch  lint',  equal  to  -  A  P,  through  which  the  arc 


PRACTICAL    CASE.  23 

B  E  is  drawn.  When  the  point  E  of  the  descril)ing  cnrve  shall  have  become  a  point  of  contact, 
as  at  E',  the  arc  E'  P  being  equal  to  E  P,  the  })oint  P  will  have  moved  to  T,  the  chord  T  E'  being 
equal  to  the  chord  E  P.  T  will  be  a  point  in  the  addendum,  and  T  E'  the  normal  for  this  point. 
From  a  point,  M ,  on  this  normal,  and  found  l)y  trial,  describe  the  arc  P  T,  limited  by  the  ad- 
dendum line.     Similarly  the  curve  of  the  dedendum  may  be  determined. 

Having  determined  such  centers  as  may  be  required  for  describing  the  tooth  curves,  draw 
circles  through  these  centers,  as  indicated  in  Plate  6,  to  facilitate  the  drawing  of  other  teeth. 
The  radius  for  the  dedendum  is  often  inconveniently  great,  and  in  such  cases  it  is  desirable  to 
use  scrolls,  employing  the  method  of  Art.  1(3,  page  11.  Next  divide  the  pitch  circle  into  as 
many  equal  parts  as  there  are  teeth,  beginning  at  the  pitch  point.  From  each  of  these  divisions 
lay  off  the  thickness  of  the  teeth.  If  there  be  no  backlash,  this  thickness  will  equal  one-half 
the  circular  pitch ;   but  if   an  amount  be  determined  for  backlash,  the  thickness  will  equal 

P' —  backlash 
2 

The  circle  of  centers  having  been  drawn,  the  tooth  curves  sliould  be  described.  These  will 
be  limited  l)y  the  addendum  and  dedendum  circles  already  drawn.     Finally  draw  the  fillets. 

The  maximum  angle  of  pressure  between  the  pinion  and  gear  will  be  24°,  the  arc  of 
approach  .52,  the  arc  of  recess  .48,  and  tlieir  sum,  which  is  the  arc  of  contact,  1  inch,  or  1.27 
times  the  circular  pitch. 

The  rack  teeth  would  be  similarly  described.  The  pitch  line  being  a  right  line,  the  circular 
pitch  may  be  laid  off  directly  by  scale,  or  spaced  from  the  pinion.  The  approximate  method 
may  be  used  for  the  tooth  curves,  and  lines  drawn  parallel  to  the  pitch  line,  for  the  centers  of 
the  arcs  which  ajjproximate  the  addenda  and  dedenda  of  the  teeth. 


24 


GEAR    FACE. 


Fig.  14. 


Fig.  15. 


35.  Face  of  Gear.  In  the  previous  consideration  of 
gear  teeth  no  attention  has  been  paid  to  the  width  of  the 
gear,  or,  as  it  is  commonly  termed,  the  face  of  the  gear. 
Tliis  dimension  is  one  of  the  factors  to  be  considered  in 
determining  the  strength  of  the  tooth,  wliich  is  a  subject 
apart  from  the  kinematics  of  gearing.  It  should  be  ob- 
served, however,  that  the  tooth  having  appreciable  width, 
nuist  be  generated  by  an  element  of  a  rolling  cylinder  in 
place  of  the  point  of  a  rolling  circle. 

36.  Comparison  of  Gears,  illustrated  in  Plates  4,  5, 
and  6.  In  tlie  three  cases  previously  considered,  the  di- 
ameter of  the  pitch  circles  are  equal,  and  only  one  diam- 
eter of  rolling  circle  has  been  used. 

In  Plates  4  and  5  the  arc  of  contact  is  equal  to  the 
circular  pitch ;  1)ut  the  pitch  of  the  latter  is  twice  as  great 
as  tlie  former,  hence  there  are  but  half  as  many  teeth.  In 
Plate  6  the  arc  of  contact  is  made  dependent  on  the 
height  of  the  tooth,  which  is  a  standard  so  chosen  as  to 
permit  of  an  arc  of  contact  sufficiently  long  for  a  practical 
case.  But  in  Plates  4  and  5  the  height  of  the  tooth  is 
dependent  on  the  arc  of  contact,  which  latter  is  made  the 
least  possible. 


COXYENTIOXAL    KKPIIE^ENTATION    OF    SPUR    (iEAPvS.  ZO 

Till'  number  of  teeili  in  the  pinions  of  Plates  4  and  6  is  the  same;  hut  in  tlie  former  the 
action  is  only  on  one  side  of  the  pitch  point,  there  being  no  a(hlenda  to  the  teeth,  hence  the 
limited  arc  of  contact. 

In  Plates  4  and  5  there  is  contact  between  only  one  pair  of  conjugate  teeth,  save  at  the 
instant  of  beginning  and  ending  contact;  while  in  the  case  of  Plate  6,  two  pairs  of  conjugate 
teeth  may  be  in  contact  during  a  part  of  the  arc  of  contact. 

37.  Conventional  Representation  of  Spur  Gears.  In  making  drawings  of  gears,  it  is  usually 
best  to  represent  them  in  section,  as  in  Fig.  14.  This  enables  one  to  give  complete  informa- 
tion concerning  all  details  of  the  gear,  save  the  character  of  the  teeth.  If  the  latter  be  special, 
an  accurate  drawing  of  at  least  two  teeth  and  a  space  will  be  required.  Should  it  be  neces- 
sary to  represent  the  geai-s  on  the  plane  of  their  pitch  circles,  as  in  Plate  6,  they  may  be  shown 
as  in  Fig.  13,  thus  avoiding  the  representation  of  the  teeth.  Again,  if  it  be  necessary  to  show 
a  full  face  view  of  the  gears,  the  method  illustrated  in  Fig.  15  may  be  employed  to  advantage. 
This  is  simply  a  system  of  shading ;  and  no  attempt  is  made  to  represent  the  proper  number  of 
teeth,  or  to  obtain  their  projection  from  another  view. 


26  INVOLUTE    8Y8TEM. 

CHAPTER    IV. 

INVOLUTE    SYSTEM. 

38.  Theory  of  Involute  Action.  If  the  describing  curves  l)e  other  than  circles  we  shall  obtain 
odontoids  differing  in  character  from  those  already  studied  ;  but  so  long  as  both  pinion  and 
gear  are  described  by  the  same  rolling  curve,  the  velocity  ratio  will  remain  constant.  The 
class  of  odontoids  illustrated  by  Plate  7,  Fig.  1,  is  known  as  the  involute,  or  single-curve 
tooth.  This  curve  cannot  be  described  by  rolling  circles,  but  may  be  generated  by  a  special 
curve  rolling  in  contact  with  both  pitch  surfaces.*  But  as  the  curve  may  be  described  by  a 
much  more  simple  process,  the  above  statement  is  of  interest  only  as  showing  the  conformity 
of  the  curve  to  the  general  law.      (Art.  14,  page  10.) 

F  and  G,  Plate  7,  Fig.  2,  are  the  centers  of  two  disks  designed  to  revolve  about  their 
respective  axes  with  a  constant  velocity  ratio,  which  is  maintained  in  the  following  manner:  — 
Suppose  tlie  disks  to  be  connected  by  a  perfectly  flexible  and  inextensible  band,  D  C  B  A,  which 
being  wound  on  the  surface  of  one,  will  be  unwound  from  the  other,  after  the  manner  of  a 
belt,  producing  an  equal  circumferential  velocity  in  the  disks.  Conceive  a  marking  point  as 
fixed  to  the  band  at  A,  so  that  during  the  motion  from  A  to  D,  curves  may  be  described  on  the 
extensions  of  disks  1  and  2,  in  a  manner  similar  to  that  described  for  the  generating  of  the 
cycloidal  curves.      When  the  point  A,  on  tiie  band,  shall  have  moved  to  B,  the  curve  Xj  B  will 

*  For  descTlption  of  this  method,  .see  MaeCord"s  Kinematics,  page  Ui5,  Airr.  279. 


TIIK    IX VOLUTE    crRVE.  27 

have  been  described  on  the  exteii.siuii  (if  disk  2.  and  B  Aj ,  on  that  of  disk  1.  When  tlie  motion  of 
the  marking  point  shall  have  continned  to  C,  Xg  Yj  C  will  have  l)een  described  on  the  extension 
of  disk  2,  and  Ag  B^  C.  on  that  of  disk  1.  Finally,  when  the  marking  point  shall  have  reached 
D,  the  curve  Xg  ¥3  Z^  D  ^\ill  have  lieen  described  on  tlie  extension  of  disk  2.  and  Ag  B^  Cj  D  on  the 
extension  of  d'sk  1. 

If  these  curves  be  made  the  outlines  of  ffear  teeth,  and  the  former  act  aoainst  the  latter 
so  as  to  produce  motion  opposed  to  that  indicated  by  the  arrows,  a  uniform  velocity  ratio  will 
be  maintained  between  the  disks.  On  investigation,  these  curves  will  be  found  to  be  involutes, 
Ag  D  being  an  involute  of  the  periphery  of  disk  1,  and  Xg  D,  an  involute  of  disk  2.  The  curves 
may.  thei'efore,  be  descrilied  liy  tlie  method  for  drawing  an  involute  (Art.  12,  page  7),  the 
path  of  contact,  A  D,  being  spaced  off  on  the  base  circle  from  A  to  Ag,  and  the  involute  drawn 
from  Ag;  or  the  line  A  D  may  be  conceived  as  wrapped  about  the  base  circle  l)eginning  the  curve 
at  D. 

39.  Character  of  the  Curve.  Plate  7,  Fig.  1,  represents  the  involute  curve  of  Fig.  2 
incorporated  into  gear  teeth.  It  becomes  necessary  to  continue  the  line  of  the  tooth  within  the 
periphery  of  the  disk,  which  will  now  l>e  designated  as  the  base  circle,  so  as  to  admit  the 
addenda  of  engaging  teeth.      This  portion  of  the  tooth  is  made  a  radial  line. 

The  pitch  point  being  at  B,  (the  intersection  of  the  line  of  centers  and  the  line  of  action), 
the  pitch  circles  will  be  drawn  through  this  point. 

The  circles  from  wliich  tlie  involute  curves  are  described,  are  called  base 

Base  Circle  Defined.  .  ,    ,      ■        ^■  i  i  •  iji  i^it 

cirdi's.      J  heu'  diameters  liear  the  same  ratio  to  each  otiier  as  do  the  (ham- 
eters  of  the  pitch  circles. 


28  CHARACTER  OF  THE  INVOLUTE. 

The  Path  of  Contact  '^'^  ^^'^^  "^^  actioii,  OF  patli  of  coiitact,  is  ii  right  lliie  tangent  to  the'  Ija.se  cir- 

a  Right  Line.         (j[gy_   jj^  jg  ^]jg  jjj^g  followetl  b}'  the  marking  point  of  the  modeh   Plate  7,  Fig.  2. 

Since  the  path  of  contact  is  a  right  hne,  and  as  tlie  common  normals  at  the  point  of  con- 

constant  Angle  of      ^'^•^^'  i^^st  alwajs  pass  through  the  pitch  point  (Art.  14,  page  10),  it  fol- 

pressure.  lows  tluit  the  line  of  pressure,  or  angle  of  the  normals,  is  constant. 

The  action  between  the  teeth  of  the  gears  in  Fig.  1,  begins  at  A,  and  ends  at  D,  taking 
Limit  of  Action        placc  Only  bctwccn  the  points  of  tangency  of  the   line  of  action  and  l)ase 
circle.     No  involute  action  can  take  place  within  the,  base  circles. 

If  the  distance  between  the  centers  of  the  gear  be  increased  or  decreased,  the  angle  of  pres- 
sure, and  length  of  the  path  of  contact  will  be  increased  or  decreased,  but  the  involute  curve, 
which  is  dependent  on  the  diameter  of  the  base  circle  only,  will  remain  unchanged.    Hence,  any 
An  Increase  in  the  cen-  c'liaugc  iu  the  distauce  bctwccn  tliB  ccuters  of  two  involute  gears  will  not 
'^Affec\'^^hrve°ioci"°'^    chaugc  tlic  vclocity  ratio,  provided  the  arc  of  action  is  (Mpial  to  the  circular 
Ratio.  pitch.     The  case  illustrated  by  Fig.  1  is  a  limiting  one  ;  and  therefore  an  in- 

crease in  the  center  distance  would  mean  an  increase  in  the  height  of  the  tooth,  in  order  that  the 
arc  of  action  shouhl  e(|ual  tlie  increased  [)itch,  an  increase  in  the  center  distance  necessitating 
an  increase  in  tlie  diameters  of  the  pitch  circles,   and  therefore  in  the  circular  pitch.      But 
wliile  the  action  between  the  teeth  continued,  the  velocity  ratio  would  remain  constant. 
Since  the  angle  of  pressure  is  constant,  and  the  paths  of  the  elements  of  a  rack  tooth  are  right 
The  Involute  Rack      b'lcs,  it  folh»ws  tluit  tlic  tootli  outliue  of  au  iuvolute  rack  nnist  be  a  right 
tooth,  a  Right  Line.     ]j,j^,^  perpcudicular  to  the  angle  of  pressure.      Plate  8  illustrates  a  rack  for 
an  inxohite  gear,  having  an  angle  of  pressure  of  about  30".      (The  section  lined  portions  are 
not  involute.) 


INVOLUTE    LIMITIXO    CASE.  29 

40.  Involute  Limiting  Case.  Pi.ati-:  S.  Let  the  diameters  of  the  pitch  cireles,  the  angle 
of  pressure,  and  the  nnniljer  of  teeth,  l)e  given.  Having  drawn  the  })iteh  cireles  ahout  their 
respective  centers,  F  and  G,  ohtain  the  hase  circles  as  follows  :  — 

Through  the  pitch  point,  B,  draw  A  D.  making  an  angle  with  the  tangent  at  the  pitch  point 
equal  to  the  angle  of  pressure,  'i'his  will  l)e  the  line  of  action  ;  and  perpendiculars,  F  A  and 
D  G,  drawn  to  it  from  centers  F  and  G.  will  determine  the  radii  of  the  base  circles,  and  the 
limit  of  the  action,  or  path  of  contact,  at  A  and  D.  This  is  a  limiting  case,  in  that  the  path  of 
contact  is  a  maximum,  and  the  arc  of  contact  equal  to  the  circular  jjitch.  Next  determine  the 
point,  C,  ])}•  spacing  the  arc,  D  K  C.  e([ual  to  DA:  A  and  C  will  he  two  points  in  the  involute 
curve  of  the  base  circle,  D  K  c,  from  which  other  points  may  be  obtained.  Similarly  describe 
D  P,  the  involute  of  the  other  l)ase  circle,  just  beginning  contact  at  D.  The  height  of  the  teeth 
will  be  limited  by  the  addendum  circles  drawn  through  D  and  A,  from  centers,  F  and  G.  The 
dedendum  circles  are  made  to  admit  the  teeth  without  clearance.  The  pinion  teeth  are  pointed, 
and  the  gear  teeth  till  the  space,  having  no  backlash.  The  circular  pitch  may  be  found  by  divid- 
ing tlie  circumference  of  the  pitch  circle  into  as  many  parts  as  there  are  teeth,  or  the  teeth  ma}' 
be  spaced  on  the  base  circle.* 

The  rack  is  made  to  engage  the  pinion  in  the  following  manner  :  — ■ 

0  being  the  pitch  point  of  rack  and  pinion,  the  right  line,  0  R,  drawn  througli  this  point, 
and  tangent  to  tlie  base  circle,  will  be  the  path  of  contact  for  motion  in  the  direction  indicated 
by  the  arrow.  The  contact  will  begin  at  R  and  end  at  S,  the  latter  i)oint  being  that  of  the 
intersection  of  the  [)ath  of  contact  and  addendum  circle.  The  rack  tooth  will  be  perpendicular 
to  the  line  of  action,  R  S:  and  the  thickness  of  tooth  will  equal  that  of  the  gear  tooth,  there 
*  For  further  details  coneeniiiig  the  construction  of  this  pinion  anJ  gear,  see  Problem  4,  Page  74. 


30  EPICYCLOIDAL    EXTENSION    OF    INVOLUTE    TEETH. 

being  no  backlash  in  either  case.  The  addendum  of  the  rack  tooth  will  be  limited  by  the 
parallel  to  the  pitch  line  drawn  through  the  first  point  of  contact,  R ;  and  the  dedendum  made 
sufficiently  great  to  admit  the  pinion  tooth  without  clearance. 

41.  Epicycloidal  Extension  of  Involute  Teeth.  The  extent  of  the  involute  action  between 
the  gear  and  the  pinion  of  Plate  8  is  limited  to  the  path  D  A ;  for  while  an  increase  in  the 
height  of  the  gear  tooth  is  possible,  the  limit  of  the  engaging  involute  tooth  is  at  A,  since  no 
part  of  an  involute  curve  can  lie  within  its  own  base  circle.  It  is,  however,  entirely  feasible 
to  continue  the  contact  by  a  cycloidal  action,  in  the  following  manner :  — 

The  angle  FAB  being  a  right  angle,  the  circle  described  on  F  B  as  a  diameter  must  pass 
through  A.  This  point  may  therefore  be  considered  as  a  point  in  an  epicycloid,  described  by 
the  rolling  circle  FAB,  and  having  A  B  for  its  normal,  which  is  also  the  normal  for  the  involute. 
I)ut  this  diameter  of  rolling  circle  being  one-half  the  pitch  circle  within  which  it  rolls,  the 
hypocycloid  will  be  a  radial  line,  and  the  dedenda  of  the  teeth  wall  be  radial  within  the  base 
circle.  By  rolling  the  same  circle  on  the  outside  of  the  gear  pitch  circle,  the  addenda  of  the 
gear  teeth  may  be  extended,  and  the  path  of  contact  continued  to  N,  which  is  a  limit  in  this 
case,  by  reason  of  the  gear  tooth  having  become  pointed. 

Similarly,  the  addendum  of  the  rack  tooth  may  be  extended  l)v  the  same  describing  circle. 
In  the  figure  it  is  made  sufficiently  long  to  just  clear  the  dedendum  circle  required  for  the 
pointed  gear  tooth.  The  action  will  now  begin  at  Q,  follow  the  I'oUing  circle  to  R,  and  then, 
becoming  involute,  continue  to  S. 

42.  Involute  Practical  Case,  Plates  9  and  10.  Having  given  the  number  of  teeth  of 
engaging  gears,  and  the  diameters  of  their  pitch  circles,  it  is  required  to  determine  the  curves 
for  the  involute  teeth  of  a  pinion,  gear,  and  rack. 


INVOLl'TE    PHACTICAL    CASE.  31 

The  diameters  of  the  (lescrihing  circles  would  be  of  lii-st  consideration  in  cycloidal  gearing  ; 
while  in  the  involute  system,  the  angle  of  pressure  or  line  of  action  nnist  first  be  established; 
and  tangent  to  this  the  base  circles  may  be  drawn.  By  reference  to  Plate  7,  Fig.  1,  it  will 
be  seen  that  with  a  constant  center  distance,  a  decrease  in  the  angle  of  pressure  will  necessi- 
tate an  increase  in  the  diameter  of  the  base  circles,  and  a  corresponding  decrease  in  the  path  of 
contact.  That  is  to  say,  an  increase  in  the  possible  length  of  the  path  of  contact  means  an 
increase  in  the  angle  of  pressure.  In  Plate  7,  Fig.  1,  this  angle  is  too  great  for  actual  prac- 
tice, being  about  30°;  yet  it  cannot  be  lessened  in  this  case,  as  the  number  of  teeth  is  limited. 
Practice  lias  limited  this  angle  to  14^°  or  15°,  which  is,  unfortunately,  too  small;  but  as  one 
of  these  angles  is  generally  adopted  in  the  manufacture  of  gears,  the  latter  will  be  used  in  the 
follo\\ing  problem  :  — 

A  pinion  of  12  teeth  is  required  to  engage  a  gear  of  30  teeth,  and  a  rack,  the  diameter 
pitch  being  1.     The  former  is  illustrated  by  Plate  9,  and  the  latter  by  Plate  10. 

Pinion  diameter  =  ci'  =  -=-=l2  inches. 

Gear  diameter     =  D'  =  ^  =  ?^  =  30  inches.     (Airr.  18,  page  11.) 

Since  the  teeth  are  to  be  of  standaid  dimensions  (Art.  31,  page  17),  the  addenda  will  Ije 
1  inch,  the  dedenda  li  inches;    and  there  being  no  backlash,  the  thickness  of  the  teeth  will  be 

half  the  circular  pitch,  or  ^.     The   circular  pitch,   P',  =p  =  3.1416.     Draw  tlie  pitch  circles. 

The  line  of  action  will  pass  through  the  pitch  point,  making  tiie  recpiired  angle  with  the 
common  tangent  at  this  point.  Next  draw  the  base  circles  tangent  to  tliis  line,  and  determine 
the  points  of  tangeucy,  D  and  A.     Construct  the  involutes  of  these  base  circles  in  the  manner 


OZ 


INTERFEREXCE. 


Fig.  16, 


indicated  by  Fig.  l(i,  and  according  to  the  nietliod  for  descril)- 
ing  an  involnte,  Ai;t.  12,  page  7.  It  will  now  l)e  seen  that 
the  gear  tooth  will  be  limited  by  the  arc  drawn  tlirougli  D, 
the  point  of  tangency  of  liase  circle  and  line  of  action.  If, 
however,  the  involnte  cnrve  be  continued  to  the  addendum 
circle,  as  shown  by  the  dotted  line,  C  E,  it  will  interfere  with 
the  radial  portion  of  the  pinion  flank,  which  lies  within  the 
base  circle.  The  pinion  tooth  will  have  no  such  limitation, 
since  the  addendum  circle  intersects  the  line  of  action,  D  A, 
at  L,  a  considerable  distance  from  the  limit  of  involute  action, 
at  the  point  A. 

Similarly,  the  rack  tooth  will  be  found  to  interfere  with 
the  pinion  flank,  if  extended  beyond  the  point  C,  Mhich 
comes  into  contact  at  the  point  D,  the  limit  of  involute  ac- 
tion. But  the  pinion  face  may  l)e  extended  indefinitely,  so 
far  as  involute  action  is  concerned.  The  remedy  for  this 
interference  is  treated  of  in  the  following  article. 

43.  Interference.  Since  practical  considerations  demand 
the  maintenance  of  a  standard  proportion  of  tooth,  two 
schemes  are  adopted  for  avoiding  or  correcting  this  inter- 
ference, observed  in  Plates  9  and  10. 

The  first  is  to  hollow  that  j)art  of  the  pinion  flank  lying 


INFLUENCE    OF    THE    ANGLE    OF    PRESSTRE.  33 

within  the  base  circle  so  as  to  clear  the  interfering  part  of  the  gear,  or  rack  tooth.  In  this 
case  there  will  be  no  action  beyond  the  point  of  tangency  D.  The  second  method  consists  in 
making  the  interfering  portion  of  the  addendum  an  epicycloid  descril)ed  by  a  circle  of  a  diam- 
eter equal  to  the  radius  of  the  pinion  pitch  circle.  Such  a  describing  circle  would  generate  a 
radial  flank  for  that  part  of  the  curve  lying  within  the  base  circle.  By  this  means,  the  action 
will  be  continued  and  the  velocity  ratio  maintained,  although  the  action  will  cease  to  be 
involute.     AuT.  41,  i)age  30. 

44.  Influence  of  the  Angle  of  Pressure.  The  interference  may  be  entirely  obviated  by 
sufificiently  increasing  the  angle  of  pressure  ;  but  in  the  case  cited  (Plates  9  and  10)  it  would 
necessitate  an  angle  of  24.1°,  which  is  too  great  for  general  use.  Had  the  number  of  teeth  in 
the  pinion  been  greater,  the  interference  would  have  been  less,  and  with  30  teeth  in  the  pinion, 
there  would  have  l)een  no  interference.     See  Art.  45. 

The  angles  of  14 i°  and  15°,  commonly  adopted,  are  unfortunately  small.  There  is,  how- 
ever, a  tendency  to  increase  this  angle,  and  gears  for  special  machines  have  been  made  with  a 
20°  angle  of  pressure.  This  latter  angle  will  permit  gears  having  18  teeth  to  engage  without 
interference,  and  the  thrust  due  to  this  increase  in  the  angle  of  pressure  is  an  insignificant 
amount.  A  system  based  on  this  angle  of  pressure  would  unquestionably  be  an  improvement 
over  the  present  one. 

45.  Method  for  determining  the  Least  Angle  of  Pressure  for  a  Given  Number  of  Teeth  having 
no  Interference.      Fig.  17. 

Let  A  be  the  center  of  a  gear  having  A  B  =^  R   foi'  the   radius   of  pitch  circle,  and  D  B  T  the 


34 


LEAST    ANGLE    OF    PRKSSURE    WITHOUT    INTERFERENCE. 


Fig.   17. 


angle  of  pressure  to  l)e  (leterinined,  the  least 
number  of  teeth  l)eing  N.  Suppose  the  gear  to 
engage  a  rack  having  standard  teeth,  then  will 


C  = 


I        D       2R 


-  =  — -  .    D  will  be  the  last  point  of  con- 

P       N         N  ^ 

tact,  and  A  D  =  r,  the  radius  of  the  base  circle. 


A  C  =  A 


C-R      p-R       N   ~        N 


A  C:A   D::A   D:A   B,  hence,  A    D2  =  r2  =  A   BxAC  = 
R'-2(N  — 2)  1  ^      /N-2 

The  angle  of  pressure,  D  B  T,  is  equal  to  an- 
gle D  A  B  =  p  ,  and 


the  cos.  p  =  1  = 
R 


V      N 


=  y/NZ2 

V      N 


Hence    the 


COS.  of  the  angle  of  pressure 


\/¥<''>- 


By  substituting  in  the  above  formula,  it  will 
be  seen  that  for  a  12-toothed  gear  to  engage  a 
rack  without  interference,  the  angle  of  action 
must  be  24.1°,  and  for  15  teeth  the  angle  would 
be   21.4°.     Again,   if   the    angle  be  15°,   the  least 


DEFE(  T>s    OF    THE    INVoLI'TE    SY.STEM.  OO 

iiuinbei-  of  teeth  that  will  engage  without  interference  will  be  80,  while  with  a  20"  aiigie  of 
pressure  the  least  number  would  be  18. 

46.  Defects  of  a  System  of  Involute  Gearing.  As  in  the  case  of  the  cycloidal  system,  it 
is  desirable  to  make  all  involute  gears  having  the  same  pitch  to  engage  correctly.  In  cycloidal 
ge'dYs  this  was  attained  by  the  use  of  one  diameter  of  rolling  circle  for  all  gears  of  the  same 
pitch  (AiiT.  33,  page  20).  In  the  involute  system  we  assume  an  angle  of  obliquity,  or  pressure, 
which  is  constant  for  all  geai-s ;  but  unless  this  angle  be  great,  gears  having  so  few  as  12 
teeth  cannot  be  run  together  without  interference.  To  obviate  this  difticulty  we  must  adopt 
one  of  the  two  methods  already  described  (Art.  43,  page  32)  ;  namely,  the  undercutting  of  the 
interfering  flanks,  or  the  rounding  of  the  interfering  addenda.  P^irst  consider  the  hitter,  which 
is  illustrated  l)y  Plates  9  and  10.  We  have  seen  how  that  portion  of  the  gear  tooth  adden- 
dum lying  beyond  the  point  C  must  be  made  epicycloidal  in  order  to  engage  the  radial  part 
of  the  pinion  flank  which  lies  within  the  base  circle ;  also  that  the  pinion  addenda  might 
be  wholly  involute  since  there  would  be  nt)  interference  with  the  gear  tooth  flank,  the  action 
between  the  latter  taking  place  without  the  base  circle.  But  if  a  12-toothed  gear  be  taken  as 
the  base  of  the  system,  it  will  be  necessary  to  round,  or  epicycloidally  extend  that  portion  of 
the  pinion  addendum  lying  beyond  the  point  K,  since  this  would  be  the  last  point  of  involute 
action  between  two  12-toothed  geai-s.  Tlierefore  when  the  12-toothed  gear  engages  one  having 
a  greater  number  of  teeth,  that  part  of  the  addendum  lying  beyond  this  point  will  no  longer 
engage  the  second  gear,  and  the  arc  of  contact  will  be  greatly  reduced.  Again,  suppose  a  pair 
of  30-toothed  gears  to  engage  (each  being  designed  to  engage  a  12-toothed  pinion),  the  only 
part  of  the  tooth  suitable  for  transmitting  a  uniform  motion  is  that  lying  between  the  base 


36 


UNSYMMETRICAL    TEETH. 


Pig.  18. 


circle  and  point  C,  Plate  D,  and  the  arc  of  contact  would 
be  but  1.05  of  the  circular  pitch.  Now,  one  of  the  claims 
made  for  the  involute  tooth  is  that  the  distance  between 
the  centers  of  the  gears  may  be  changed  without  changing 
the  velocity  ratio  ;  but  in  this  latter  case  it  cannot  be  done 
without  making  the  arc  of  contact  less  than  the  circular 
pitch. 

If  the  system  of  midercutting  the  flanks  be  adopted, 
the  addendum  will  be  wholly  involute;  and  in  the  case  of 
Plate  9  all  of  the  })inion  addendum  would  have  been 
available  for  action,  Init  the  pinion  flaidv,  within  the  base 
circle  would  have  been  cut  away  so  that  there  would  have 
been  no  action  of  the  gear  addendum  between  C  and  E. 
If,  however,  the  engagement  had  been  between  two  30- 
toothed  gears,  all  of  the  tooth  would  have  been  available 
for  action,  and  the  arc  of  contact  would  have  been  equal 
to  1.91  of  the  circular  pitch. 

Thus  it  will  be  seen  that  involute  gears  should  be  de- 
signed to  engage  the  gears  with  which  they  are  intended 
to  run,  if  the  best  results  would  be  attained.  This  would, 
of  course,  prevent  the  use  of  the  ready-made  gear  or  cut- 
ter, but  would  insure  a  longer  arc  of  action  between  con- 
jugate teeth. 


UNSYMMETinCAL    TKETH.  37 

47.  Unsymmetrical  Teeth.  Fiy-.  IS.  A  very  dcsiialiK',  altliou^li  little  used,  form  of  tooth 
is  that  known  as  the  inisynimetrieal  tooth,  which  usually  combines  the  cycloidal  and  involute 
systems.  Fig.  18  illustrates  a  pinion  and  gear  haying  the  same  number  of  teeth  as  those  illus- 
trated l)y  Plate  4,  and  the  arc  of  contact  is  unchanged  ;  but  the  angle  of  pressure  is  much 
reduced,  and  the  strength  of  the  tooth  increased.  As  the  involute  face  of  the  tooth  is  designed 
to  act  only  when  it  may  be  necessary  to  reverse  the  gears,  and  when  less  force  would  usually  be 
transmitted,  the  angle  of  pressure  may  be  made  greater  than  ordinary.  In  this  case  the  angle 
is  24.1°,  which  is  sufficient  to  avoid  interference  in  a  standard  12-toothed  gear  (Art.  45, 
page  33).  But  this  angle  is  no  greater  than  the  maximum  angle  of  pressure  in  Plate  4. 
This  reinforcement  of  the  \ydck  of  the  tooth  makes  it  possible  to  use  a  much  greater  diameter 
of  rolling  circle  ;  and  in  the  case  illustrated,  the  diameter  is  one-third  greater  than  the  radius 
of  the  pitch  circle.  Tliis  increase  in  the  diameter  of  tlie  rolling  circle  would  have  lengthened 
the  arc  of  contact,  had  not  the  height  of  the  tooth  been  reduced  to  maintain  the  same  ai'c  as 
that  of  Plate  4. 

The  cycloidal  action  begins  at  C  and  cuds  at  H,  making  a  maxinnuu  angle  of  pressure  of 
17".  The  same  i-oUing  circle  has  been  ust'd  for  the  face  and  flank  of  each  gear;  Imt  the  one 
rolling  within  the  pitch  circle  of  the  gear  might  have  been  nmch  increased  without  materially 
weakening  the  gear  tooth. 

The  involute  action  would  ))egin  at  D  and  end  at  B,  making  an  arc  of  contact  a  little  greater 
than  the  pitch. 


OO  ANNULAR    GP:AR1NG. 

CHAPTER    V. 

ANNULAR     GEARING. 

48.  Cycloidal  System  of  Annular  Gearing.  If  the  center  of  the  pinion  lies  within  the  pitch 
circle  of  the  gear,  the  latter  is  called  an  internal,  or  annular  gear.  The  solution  of  problems 
relating  to  this  form  of  gearing  differs  in  no  wise  from  that  of  the  ordinary  external  spur  gear, 
save  in  the  consideration  of  certain  limitations  which  will  be  treated  of. 

49.  Limiting  Case.  Plate  11  illustrates  a  pinion  engaging  an  internal  and  an  external 
spur  geai'.  The  pinion  has  6  teeth,  and  the  gears  have  13  teeth.  The  arc  of  contact  is  made 
equal  to  the  circular  pitch,  and  equally  divided  between  recess  and  approach.  The  pinion  has 
radial  flanks,  which  therefore  determines  the  diameter  of  the  describing  circle  for  the  addenda 
of  the  gears.  The  second  describing  circle,  2,  is  governed  by  conditions  which  will  appear  later. 
It  will  be  observed  that  the  addenda  of  the  annular  gear  teeth  lie  within,  and  the  dedenda 
without,  the  pitch  circle.  The  height  of  the  teeth  is  governed  by  the  arcs  of  approach  and 
recess ;  and  the  construction  of  the  teeth  does  not  differ  from  the  limiting  case  considered  in 
Art.  26,  page  14,  and  Plate  5.  The  action  between  the  pinion  and  annular  gear  begins  at 
B,  and  ends  at  C,  the  pinion  driving. 

50.  Secondary  Action  in  Annular  Gearing.  We  have  already  seen.  Art.  10,  page  (>,  that 
every  epicycloid  may  be  generated  by  either  of  two  rolling  circles,  which  differ  in  diameter  by 


SECONDARY    ACTION    IN    ANNULAR    GEARING.  39 

an  amount  equal  to  the  diameter  of  the  pitch  circle.  Also,  that  every  hypocycloid  may  he 
generated  by  either  of  two  rolling  circles,  the  sum  of  the  diameters  of  which  shall  equal  that 
of  the  pitch  circle  within  which  they  roll.  Thus  the  addendum,  C  E,  of  the  pinion,  Plate  11, 
may  he  described  by  the  circle  2,  or  the  intermediate  circle  3.  But  in  this  case  the  circles  1 
and  2  are  so  chosen  that  the  intermediate  circle  3  is  the  second  describing  circle  for  the  hypo- 
cycloid  F  G ,  as  well  as  for  the  epicycloid  C  E ;  consequently  C  E  and  F  G  will  produce  a  uniform 
velocity  ratio,  the  contact  taking  place  from  A  to  D.  The  addendum  C  E  has  contact  also  with 
the  dedendum  C  F  along  the  path  A  C ;  hence,  during  a  part  of  the  arc  of  recess  there  must  be 
two  points  of  each  tooth  in  contact  at  the  same  time. 

The  plate  illustrates  the  contact  along  the  path  A  C  as  just  completed  ;  but  a  second  point 
of  contact  will  be  seen  on  circle  3,  between  F  and  E,  and  action  along  this  path  will  be  con- 
tinued to  D.  The  case  is  therefore  no  longer  a  limiting  one,  inasmuch  as  the  arc  of  contact  is 
greater  than  the  circular  pitch.  'J'he  additional  contact  takes  place  during  the  arc  of  recess, 
which  is  also  advantageous. 

In  order  to  ol)tain  this  secondary  action,  the  sum  of  the  radii  of  the  inner  and  outer  rolling 
circles  must  equal  the  distance  hetween  the  centers  of  pinion  and  (/ear.* 

For,  letting  r,,  r.^,  and  r:;  be  the  radii  of  the  inner,  outer,  and  intermediate  rolling  circles, 
and  Rp,  Rg,  the  radii  of  pinion  and  gear,  rg  +  r^  =  Rg,  (^6),  and  i-g  —  r.^  =  Rp,  (7),  Art.  10,  page  (3. 
Subtracting  the  second  equation  from  the  fii'st,  rj  +  r.,  =  Rg  —  Rp  ='C  =  center  distance  (8). 

Plate  12,  Fig.  1,  illustrates  the  same  pinion  and  gear,  the  teeth  having  been  described  by 
the  intermediate  circle  only.      In  this  case  the  action  takes  place  wholly  during  recess,  the  arc 

*  The  student  is  referred  to  Prof.  MacCord's  "Kinematics,"  images  104  to  10!)  inclusive,  for  a  very  complete 
demonstration  of  this  law,  together  with  other  limitations  of  aniudar  gears. 


40  LIMITATION    OF    INTERMEDIATE    DESCRIBING    CURVE. 

of  recess  being  the  same  as  before,  about  1  j  times  the  cireuhir  pitch.  Had  the  outer  describing 
circle  been  used  to  describe  the  dedenda  of  the  gear  teeth,  as  in  the  })reeeding  cases,  a  secondary 
action  wouhl  have  taken  place  during  the  recess. 

Special  notice  should  be  taken  of  the  reduced  angle  of  pressure  in  the  secondarv  action  of 
annular  gearing,  and  of  the  possibility  of  obtaining  a  great  arc  of  recess  \vith  little  or  no 
approacliing  action.  These  advantages  are  very  apparent  in  Plate  11,  in  which  the  pinion 
engages  an  external  and  an  internal  gear  having  an  equal  numl)er  of  teeth. 

51.  Limitations  of  the  Intermediate  Describing  Circle.  Plate  12,  P'ig.  2,  Suppose  the 
inner  describing  circle,  1,  Plate  11,  to  l)e  increased  luitil  it  equals  the  diameter  of  the  pinion 
pitch  circle,  Of,  the  radius  of  the  intermediate  describing  circle  will  then  equal  the  center  dis- 
tance, 5}^,  and  the  outer  describing  circle,  2,  would  be  but  \'^  radius.  For  by  sul)stituting 
Rp  for  rj  in  equations  6  and  8,  Ai;t.  50,  we  shall  obtain  r,  =  Rg  —  Rp  =  c,  and  r,  =  C  —  Rp.  Plate 
12,  Fig.  2,  illustrates  this  case,  the  outer  describing  circle  not  being  employed. 

Since  the  pinion  pitch  circle  has  now  become  a  describing  curve,  there  will  l)e  an  appi'oach- 
ing  action  ;  Ijut  only  one  point  of  the  pinion  tooth  will  act,  as  the  diameter  of  the  describing 
circle  and  pitch  circle  being  equal  reduces  the  pinion  flank  to  a  point.  But  if  any  further 
increase  be  made  in  the  diameter  of  the  inner  circle,  whicli  is  equivalent  to  a  decrease  in  the 
intermediate  describing  curve,  an  interference  will  take  place  during  approaching  action  ;  since 
the  curves  of  gear  and  pinion  teeth,  generated  by  a  circle  greater  than  the  pinion  diameter,  will 
cross  one  another,  which  would  make  action  impossible.  Hence,  the  radius  of  the  intermediate 
describing  circle  ca)uiot  be  less  than  the  line  of  centers. 

52.  Limitations   of    Exterior  and   Interior  Describing  Circles.      Plate  12,   Fig.  8.      From 


LIMITATION    OF    EXTEIJIOR    AND    INTERIOR    DESCRIBING    CURVES.  41 

Art.  50,  page  39,  it  was  seen  that  the  sum  of  the  radii  of  tlie  exterior  aiul  interior  descril)ing 
circles  must  equal  the  center  distance  if  a  secondary  action  be  obtained.  If  either  circle  be 
decreased  without  decreasing  the  other,  the  secondary  action  ceases ;  but  if  either  circle  be 
increased  wdthout  an  equal  decrease  in  the  other,  thus  making  the  sum  of  their  radii  greater 
than  the  center  distance,  the  addenda  will  interfere.  Thus,  iii  Plate  11,  a  decrease  in  describ- 
ing circle  2  would  produt-e  a  more  rounding  face,  and  C  E  would  fail  to  engage  F  G;  but  had 
this  describing  circle  been  increased  in  diameter  without  a  corresponding  decrease  in  1,  C  E 
would  have  interfered  with  F  G  .  Hence,  tlie  limit  of  the  sum  of  the  radii  of  the  exterior  and 
interior  describing  circles  is  the  center  distance, 

Plate  12,  Fig.  •],  illustrates  a  special  case  of  the  above  condition,  the  interior  describing 
circle  being  reduced  to  zero,  and  the  radius  of  tlie  exterior  circle  made  equal  to  the  center  dis- 
tance, thus  making  the  intermediate  describing  circle  equal  to  the  [)itch  circle  of  the  gear. 
There  will  be  double  contact  during  a  portion  of  the  arc  of  recess,  the  contact  l^eginning  at  A, 
and  following  the  outer  describing  circle  to  C,  and  the  intermediate  (or  in  this  case  the  pitch 
circle  of  the  gear)  to  D.  Tliis  design  is  ol)jectionable  in  that  the  secondary  action  takes  place 
with  only  one  point  of  the  gear  tooth. 

53.  The  Limiting  Values  of  the  Exterior,  Interior,  and  Intermediate  Describing  Circles  for 
Secondary  Action.  !Since  r.^H-  ri  =  C,  either  radius  will  equal  C,  when  the  other  becomes  zero; 
l)ut  if  there  l)e  a  secondary  action,  the  minimum  value  of  r.,  may  not  be  zero,  for  r^  will  be  a 
maximum  when  r.,  is  a  mininnim.  as  rj  +  rs  =  Rg,  r-  is  a  minimum  when  equal  to  C  (Art.  52), 
and  substituting  this  value  in  the  last  equation,  rj  =  Rg  —  C  .  Again  suhstituting  this  value  in 
the  equation,  r._,  +  rj  =  C ,  r.  =  C  —  (Rg  —  C)  =  2 C  —  Rg. 


42  LIMITING    VALUES    OF    DESCIUBING    CIRCLES    FOR    SECONDARY    ACTION. 

Summary  of  the  above  limiting  values  and  conditions  governing  secondary  action:  — 

rj  maximum  =  Rg  —  C  ;  r^  minimum  =  0  ;  rs  +  rj  =  Rg  .        (G) 

r.^  maximum  =  C  ;  r^  minimum  =  2  C  -  Rg  ;  rs  —  r^  =  Rp .        (7) 

rg  maxinmm  =  Rg ;  h  mininuun  =  C  ;  Rg  -  Rp  =  C  .      (8) 

54.  Practical  Case.  If  annular  gears  be  made  interchangeable  with  spur  gears,  it  will  be 
necessary  to  have  the  number  of  teeth  in  the  engaging  gears  differ  b}-  a  certain  number  which 
will  depend  on  the  base  of  the  system.  This  is  due  to  the  limitation  in  the  sum  of  the  radii  of 
the  describing  circles,  Art.  52,  page  40.  Thus,  let  12  be  the  base  of  the  system,  and  it  is 
required  to  find  the  least  number  of  teeth  in  the  annular  gear  that  will  engage  the  pinion.  If 
the  pitch  be  2,  the  diameter  of  the  pinion  will  be  6,  and  that  of  the  describing  circles  8.  But 
since  the  center  distance  cannot  be  greater  than  the  sum  of  the  radii  of  the  describing  circles 
(in  this  case  3),  the  diameter  of  the  annular  gear  must  be  12,  and  the  least  number  of  teeth  in 
the  annular  gear  will  be  24. 

Using  the  notation  of  Plate  11,  and  Art.  17,  page  11,  n  being  the  least  number  of  teeth 
in  the  gear,  and  n  the  least  number  in  the'  pinion,  or  the  base  of  the  system:  — 

n  1         „        „  r^  N  n       1  n  N  n  ^  ., 

C  =  2  r^  =  —  ,  also  C=Rg-Rp=  —  -  — ,  hence  ^  =  ^  -  ^^  or  2  n  =  N  . 
The  least  number  of  teeth  in  the  annular  gear  will  be  twice  that  of  the  base  of  the  system. 

55.  Summary  of  Limitations  and  Practical  Considerations,  (a)  The  diameter  of  the  inter- 
mediate describing  circle  is  equal  to  the  diameter  of  the  pinion,  plus  the  diameter  of  exterior 
describing  circle,  or  diameter  of  gear  minus  interior  describing  circle.      (Art.  10,  page  6.) 


SUMMARY    OF    LIMITATIONS    AND    PRACTICAL    CONSIDERATIONS.  43 

(^)  There  will  be  .secondaiy  action  only  when  the  sum  of  the  radii  of  the  exterior  and 
interior  deseril)ing  cireles  is  e(|ual  to  tlie  line  of  centers.      (Art.  50,  page  38.) 

((?)  The  radius  of  the  intermediate  descril)ing  circle  cannot  be  less  than  the  center  distance. 
(Art.  51,  page  40.) 

((?)  The  sum  of  the  radii  of  exterior  an.d  interior  describing  circles  cannot  be  greater  than 
the  center  distance.     (Art.  52,  page  40.) 

(^)  The  nundjer  of  teeth  in  any  pair  of  gears  of  an  interchangeable  system  must  differ  by 
an  amount  equal  to  the  ])ase  of  the  system.      (Art.  54,  page  42.) 

(/)  If  the  pinion  drives,  the  exterior  describing  circle  should  be  the  greater  in  order  tliat 
the  arc  of  contact  may  be  chiefly  one  of  recess. 

(</)  If  the  gear  drives,  the  interior  describing  circle  should  be  the  greater,  and  the  pinion 
teeth  may  have  flanks  only,  but  in  this  case  the  teeth  should  be  extended  slightly  beyond  the 
pitch  circle  in  order  to  protect  the  last  point  of  contact,  which  will  be  on  the  pitch  circle. 

56.  Involute  System  of  Annular  Gearing.  Fig.  11).  The  method  of  drawing  the  tooth 
outlines  for  the  involute  annular  gear  does  not  differ  from  that  of  the  spur  gear.  Pitch  lines 
having  been  determined,  the  base  circles  are  drawn  tangent  to  the  line  of  action,  and  the  invo- 
lutes of  those  base  circles  will  be  the  required  curves.  Care  must  be  used  in  obtaining  the 
length  of  the  teeth,  in  order  to  avoid  a  second  engagement  after  the  full  action  shall  have 
taken  place.  To  determine  if  this  interference  takes  place,  it  is  necessary  to  construct  the 
epitrochoid  of  the  point  of  the  pinion  tooth,  or  determine  the  path  of  least  clearance,  as  in 
Art.  28,  page  15. 


44 


INVOLUTK    8Y8TKM    OF    ANNULAR    (iEATvING. 


A'V^ 


Fiy.  1*1  illustrates  an  aiiiinlar  y;vin-  of  20  teeth  en- 
gaging a  })inion  of  10  teeth,  the  angle  of  pressure 
being  20°.  The  pinion  driving  in  the  direction  indi- 
cated will  establish  the  first  point  of  contact  at  A,  and 
the  last  point,   B,  will  lie  limited  by  the  height  of  the 


i 


tooth,  in  this  case 


P" 


The  limit  of  the  Q-ear  tooth  will 


be  determined  by  the  arc  drawn  from  the  center  of 
gear  througli.  the  point  A,  Any  extension  of  the  in- 
volute beyond  this  point  will  interfere  with  the  pinion 
flank.  The  stronger  form  of  the  annular  gear  tooth 
permits  of  a  greater  clearance,  whieh  it  is  advantageous 
to  a(h)pt. 

If  the  pillion  and  gear  differ  but  little  in  diameter, 
it  is  desirable  to  use  the  cycloidal  system,  in  ^\  Inch  case 
the  interference  may  be  more  easily  avoided.  It  should 
also  be  noted  that  the  advantages  to  be  derived  from 
an  increase  in  the  arc  of  contact  and  a  decrease  in  tlie 
angle  of  pressure  are  only  to  be  obtained  by  the  use  of 
the  latter  svstem. 


Pig.  19. 


TllKORV    OF    BEVEL    (LEAKING.  45 

CHAPTER    VT. 

BEVEL     GEARING. 

57.  Theory  of  Bevel  Gearing.  In  all  cases  previously  considered,  the  elements  of  the  teeth 
were  parallel,  the  surfaces  having  heen  generated  by  a  right  line  which  was  either  an  element 
of  a  rolling  cylinder,  as  in  the  cycloidal  system,  or  hy  an  element  of  a  flexible  band  parallel  to 
the  axis  of  a  cylinder  from  which  it  was  unwrapped,  as  in  the  involute  system.  All  sections 
of  the  teeth  made  by  planes  perpendicular  to  the  axis  were  alike,  and  therefore  it  was  only 
necessary  to  consider  one.  Under  these  conditions  the  pitch  cyclinder  became  a  pitch  circle, 
and  the  describing  cylinder  a  describing  circle.  If  we  now  consider  the  axes  of  the  gears  as 
ijitersecting,  the  friction  cylinders  will  become  friction  cones,  the  describing  cylinder  will  be  a 
describing  cone,  and  the  elements  of  the  teeth  will  converge  to  the  point  of  intersection  of  the 
axes,  making  all  sections  of  the  teeth  to  differ  from  one  another. 

Fig.  20,  page  46,  illustrates  this  case.  A  C  B  and  BCD  are  two  friction  cones,  or  pitch  cones, 
having  axes  G  C  and  H  C.  The  outlines  of  the  teeth  are  drawn  on  the  s})herical  l)ase  of  the 
cone,  that  portion  of  the  curve  lying  outside  tiie  pitch  cone  being  a  spherical  epicycloid,  and 
that  within,  a  s})herical  hypocycloid.  The  dedendum,  or  surface  of  the  tooth  lying  within  the 
])itcli  cone  A  C  B,  was  described  by  the  element  E  F  C  of  the  describing  cone,  which  is  shown  as 
generating  the  acUlendum  of  the  pinion  tooth.  Only  that  portion  of  the  surface  described  by  E  F 
would  be  used  for  the  pinion  tooth,  the  length  of  the  gear  tooth  having  been  limited  as  shown. 
The  describing  cone  employed  for  generating  the  addendum  of  gear,  and  dedendum  of  pinion, 


46 


CHARACTER  OF  CURVES  IN  BEVEL  GEARING. 


is  not  shown ;  Init  the  dia.meter  of 
its  base  would  be  governed  by  laws 
similar  to  those  already  considered 
for  limiting  the  diameters  of  rolling 
circles,  Art.  32,  page  18. 

58.  Character  of  Curves  employed 
in  Bevel  Gearing.  The  cycloidal 
BEVEL  TOOTH  has  already  been  con- 
sidered in  the  previous  article,  and 
the  curve  does  not  differ  from  that 
employed  in  spur  gearing,  save  that 
it  is  described  on  the  surface  of  a 
sphere. 

It  is  important  to  note  that  no 
tooth  can  be  made  with  a  radial  flank, 
since  no  circular  cone  can  be  made 
to  generate  a  plane  surface  by  roll- 
ing within  another  cone,  but  the 
flank  may  approximate  closely  to 
such  plane. 

The    INVOLUTE    BEVEL    TOOTH   is 

one  havino-  a  srreat  circle  for  its  line 


Pig.  20 


TKP:I)G<)L1)    AJ'lMtoXlMATlON. 


47 


of  action.  Fig.  :21  illustrates  a  crowu  gear  of  this  type.  A  C 
is  a  great  circle  of  the  sphere  A  D  C  E ,  and  is  tangent  to  the 
circles  A  E  and  DC.  If  the  circle  A  C  he  rolled  on  D  C,  so  as 
to  continue  tangent  to  D  C  and  A  E,  the  point  B  will  descrihe 
the  spherical  involute  G  B  F.  Conjugate  teeth  described  by 
this  process  maintain  their  velocity  ratio  constant,  even  while 
undergoing  a  slight  change  in  their  shaft  angles,  thus  conform- 
ing to  the  general  character  of  involute  curves. 

The  OCTOID  BEVEL  TOOTH  is  one  having  a  plane  surface 
for  the  addendum  and  dedendum,  the  plane  being  such  as 
Avould  cut  a  great  circle  from  the  surface  of  the  sphere.  In 
Fig.  22,  G  F  is  the  plane  which  cuts  the  surface  of  the  tooth 
shown  at  B .  The  line  of  action,  from  which  the  tooth  takes 
its  name,  is  indicated  l)y  the  curve  B  C  E  B  H  K  .  This  tooth 
was  the  invention  of  Hugo  lidgram,  and  is  of  interest  in  being 
the  only  bevel  tooth  that  can  be  formed  in  a  practical  manner 
by  the  molding-planing  process.  The  lUlgram  machine,  de- 
signed to  plane  this  tooth,  is  descril)ed  in  the  Journal  of  the 
Franklin  Institute  for  August,  188G,  and  in  the  American  Ma- 
chinist  for  Mny  0,  1885. 

59.  Tredgold  Approximation.  I  because  of  the  diiticulty  in- 
volved in  descril)ing  the  tooth  form  on  the  surface  of  a  sphere. 


Fig:  21. 


Fig;  22. 


DKAFTINC    THE    BEVEL    GEAll. 

it  is  custoniiiiy  to  draw'  the  outline  on  the  developed  surface  of 
a  cone   which   is    tangent   to    the   sphere   at   the   pitch   circle. 
Tliis  cone  is  called  the  normal,  or  back  cone.     Plate  13  il- 
lustrates a  sphere  A  B  D,  from  which  the  pitch  cones  A  C  B 
and  BCD  have  been  cut.      'J'angent  to  the  sphere  at  the 
pitch  circles,   A  B  and  B  D,   are   the   normal  cones  A  G  B 
and  B  H  D,  the  elements  of  which  are  perpendicular  to 
the  intersecting  elements '  of   the  pitch  cones.      The 
error  in  the  tooth  curve  due  to  this  apjjroximation 
is  so  small  as  to  be  inappreciable,  save  in  exag- 
gerated  cases ;    and   the    method    is    always   em- 
ployed for  the  drafting  of  bevel  gears. 


Fig.  23. 


60.    Drafting  the  Bevel  Gear.     Plate 
13,   and   Fig.   23.      The    drawing    usually 
required   is  that  illustrated  by  Fig.  23, 
which  is   a  section   of  a  gear  and  pin- 
ion, together  with  the   development  of 
a   portion   of    the  outer  and  inner  nor- 
mal cones,  only  the  tooth  curves  bt-ing 
^o^   omitted. 

The  names  of  the   parts   of   a  bevel 
gear  are    also  given,   and    the  lettering 


DRA FATING    THE    BEVEL    GEAR.  40 

corresponds  to  that  of  Plate  1-5,  Avliicli  latter  will  be  used  to  illustrate  the  method  of 
draAvi  ng. 

A  B  and  B  D,  Plate  13,  are  the  pitch  diameters  of  a  gear  and  pinion  with  axes  at  90"^,  and 
liaving  15  and  12  teeth  resi)ectively,  the  pitch  being  3,  when  drawn  to  the  scale  indicated. 
The  pitch  diameters  being  5"  and  4",  lay  off  C  K  on  the  center  line  of  gear,  equal  to  one-half 
the  pitch  diameter  of  pinion,  and  C  L  on  the  center  line  of  pinion,  equal  to  one-half  the  pitch 
diameter  of  the  gear.  Through  these  points  draw  the  pitch  lines  perpendicular  to  the  axes  of 
the  g-ears,  and  in  this  case  perpendicular  to  each  other.  Draw  the  pitch  cones  A  C  B  and  BCD, 
and  perpendiculai'  to  these  elements  draw  G  A,  G  B  H,  and  H  D,  elements  of  the  normal  cones. 
Having  figured  the  addendum  and  dedendum  of  the  teeth,  la}'  off  on  the  normal  cone  of  pinion 
B  M  and  B  N ,  D  0  and  D  Q ,  and  from  these  points  draw  lines  converging  to  the  apex  of  the  pitch 
cones.  Similarly  lay  off  addenda  and  dedenda  of  gear,  limiting  the  length  of  the  face  at  R  by 
drawing  the  elements  of  the  inner  normal  cones  at  R  S  and  R  T.  The  face  B  R  should  not  be 
gi'eater  than  one-third  B  C,  by  reason  of  the  objectionable  reduction  in  small  end  of  teeth. 
Complete  the  gear  blank,  or  outline,  by  drawing  the  lines  limiting  the  thickness  of  the  gear, 
diameter  and  length  of  hub,  diameter  of  shaft,  etc.,  details  which  are  matters  of  design. 

The  development  of  the  normal  cone  of  the  gear,  B  G  A,  will  be  a  circular  segment  described 
with  radius  G  B,  and  equal  in  length  to  the  circumference  of  the  pitch  circle  of  the  gear. 
Since  there  are  15  teeth  in  the  gear,  the  developed  pitch  circle  will  be  divided  into  15  parts, 
as  shown,  and  the  circular  'pitch  be  thus  determined.  But  it  is  unnecessary  to  obtain  the 
complete  development  as  shown  in  the  plate,  since  the  shape  of  one  tooth  and  space  is  alone 
required.  Therefore,  space  oft'  on  a  portion  of  the  arc  of  the  developed  2')itch  circle,  the  cir- 
cular pitch,  B  V,  A\hich  is  equal  to  p.      Draw  the  addendum  and  dedendum  circles  with  radii 


50  DRAFTING  THE  BEVEL  GEAR. 

equal  to  distance  of  these  circles  from  the  a])ex  of  tlie  normal  cone,  wliicli  in  the  case  of  the 
gear  will  be  G  E  and  G  F. 

Next  determine  the  tooth  cnrve  as  for  spur  gears,  nsiiig  tlie  developed  pitch  circle  instead 
of  the  real  pitch  circle.  In  tlie  case  illnstrated,  the  cnrve  is  involute.  B  w  is  a  part  of  the 
line  of  action,  making  an  angle  of  75'^  with  G  hi,  the  line  of  centers.  The  hase  circles  drawn 
tangent  to  this  line  will  be  the  circles  from  which  the  involutes  are  described.  Had  the  cycloidal 
system  been  employed,  the  diameter  of  the  rolling  circle  would  have  been  made  dependent  on 
the  diameter  of  the  developed  pitcdi  circle,  instead  of  the  pitch  diameter  A  D. 

In  like  manner  obtain  the  development  of  the  inner  normal  cones,  having  S  R  and  T  R  for 
elements,  and  describe  the  true  curves  of  the  small  end  of  teeth.  These  pitch  circles  may  be 
drawn  concentric  with  the  developed  pitch  circles  of  the  outer  cones,  or  with  S  and  T  as  centers, 
the  latter  l)eing  the  method  commonly  adopted.  I)otli  methods  have  been  employed  in  the 
plate.  If  the  development  of  the  inner  pitch  cone  of  gear  be  di-awn  fi'om  the  center  G,  the 
reduced  pitch,  and  thickness  of  tooth,  may  be  obtained  by  drawing  the  radial  lines  from  the 
development  of  the  outer  cone  as  shown  l)y  the  fine  dotted  lines.  The  addendum  and  deden- 
dum  circles  will  lie  described  with  radii  S  Z  and  S  Y,  and  the  tooth  curves  may  be  drawn  by 
determining  the  leduced  rolling  circh%  if  the  gear  be  cycloidal,  or  the  reduced  base  circle  if  the 
involute  system  be  em])loyed. 

A  second  method  foi' describing  the  teeth  on  the  inner  normal  cone  would  1)0  to  l)asc  it  directly 
on  the  reduced  ])itch,  which  may  be  determined  by  dividing  the  nundier  of  teeth  by  the  diameter 
of  tlie  base  of  the  pitch  cone  at  this  point.     In  the  plate,  the  value  of  P  for  small  end  of  teeth 

l5  12       .  .     . 

is  _—  for  gear,  or  r-~  for  pinion  -=  4  6  ==  P.  TIk;  addendum,  dedendum,  circular  pitch,  etc.,  may 
now  be  obtained  from  this  value  of  P,  as  was  done  in  the  case  of  the  outer  pitch  cone.      In  like 


FIGURING    BEVEL    GEARS. 


51 


manner  we  may  obtain  any  otlier  section 
of  the  tooth,  although  a  third  section  is 
seldom  required. 

6i.  Figuring  the  Bevel  Gear  with  Axes 
at  90°.  Figs.  24  and  25.  The  dimensions 
required  for  the  figuring  of  a  pair  of  bevel 
gears  will  be  :  — 

First :  Those  required  for  general  refer- 
ence, and  consisting  of  pitch  diameters, 
number  of  teeth  (or  pitch),  face  (K),  thick- 
ness of  geare  (L  and  M)  (U  and  V),  diameter 
and  length  of  hubs. 

Second :  In  addition  to  the  above,  the 
pattern  maker  and  machinist  will  require, 
for  the  turning  of  the  l)l;ink.  the  outside 
diameter,  backing,  angle  of  edge,  angle  of 
face. 

Third :  The  cutting  angle  will  be  re- 
quired for  cutting  the  teeth. 

The  figures  required  for  the  fii*st  set  of 
dimensions  are  all  matters  of  design,  but 
the  second  and   third    dimensions   must  be 


I      BACKING 


52 


BEVEL    GEARIN(;. 


deteniiiiied  from  the  data  given  in  the  first.  To  ol)tain  these  it  is  neeessar}-  to  figure  the  five 
dimensions  indicated  in  Fig.  25,  three  of  which.  A,  B,  and  C,  are  angles,  and  two,  E  and  F,  are 
necessary  to  determine  the  outside  diameter  and  backing.  Only  one  of  tliese,  A,  is  used 
directly.  B  is  called  the  angle  increment,  C  the  angle  decrement,  E  is  one-half  the  diameter 
increment  of  the  pinion,  and  F  is  equal  to  one-half  the  diameter  increment  of  the  gea,r. 
In  the  similar  right  triangles  a  b  t  and  t  r  m.  Fig  25, 


ta  b=m  t  r  =  A. 


ab  =  ^. 


b  t  = 


d' 


t  m  -- 


,  d'  n 


tan   B  ^ 


d' 


2  sin   A   _  2  sin  A 
P  d'      "^        n 


E  =  -  cos  A 


2  sm  A 

The  angle  decrement,  C,  is  sometimes  made  equal  to  B,  in  which  case  the 
dedendum  of  the  tooth  at  the  small  end  will  he  greater,  as  shown  by  the  line 
h  u ;   but  if  the  bottom  line  of  the  tooth  be  made  to  converge  to  the  apex  of 
the  pitch  cones,  the  angle  t  a  h,  or  C,  will  be  determined  as  follows: 

F  f   P 

tan   L  =  — 


d' 

9 

sin 

A 

2.25 

sin 

A 

2 

4 

n 

n 

sin 

A 

Fig.  25. 


Having  determined  tliese  values,  it  is  only  necessary  to  combine  them  with  those  fixed  by 
the  design  to  complete  the  figuring  of  the  gear  as  shown  in  Fig.  24. 

Tiie  angles  should  be  expressed  in  degrees  and  tenths,  rather  than  in  degrees  and  minutes. 
It  is  also  of  importance  that  the  outside  diameter  and  backing  be  figured  in  decimals,  to  thou- 
sandths, I'ather  than  in  fractional  e(|uivalents. 


I5i:VEL    GEAR    TABLE.  53 

62.  Bevel  Gear  Table  for  Shafts  at  9o\  In  order  to  facilitate  the  figuring  of  bevel  goal's, 
tables  or  charts  of  the  principal  values  are  commonly  employed.  Such  cliarty  also  make  the 
figuring  possible  to  those  unfamiliar  with  the  solution  of  a  right  triangle.  Some  are  designed 
to  solve  the  problems  graphically,  while  others,  like  the  following,  pages  5-4  and  55,  consist  of 
the  trigonometrical  functions  for  gears  of  the  proportions  commonly  employed. 

Desckiptiox  of  Table.* 

CoLUMX  1.    Ratio  of  Pinion  to  Gear.  ,,       ^ 

Column  2.    Katio  of  Pinion  to  Gear  expressed  in  decimals,  or  tang  of  center  angle,      tan  ^  ""  r,  —  7,' 

Column  3.    Center  angle  of  Pinion  corresponding  to  tangent  in  column  2. 

Column  4.  Ten  times  the  angle  increment  for  a  Pinion  of  10  teetli.  This  increased  value  is  employed  to 
simplify  the  figuring  of  gears  having  other  than  10  teeth.  Thus,  the  angle  increment  for  miter  gears 
(1  to  1)  having  10  teeth  would  be  8.2°,  and  for  H  teeth,  {f  of  this  value  or  J 7.  There  is,  of  course,  a 
slight  error  in  deriving  the  angle  increment  for  any  number  of  teeth  from  these  values,  in  that  the 
tangent  and  arc  do  not  vary  alike,  but  the  error  is  inappreciable  for  small  arcs. 

Column  5.    The  diameter  increment  for  a  Pinion  of  one  pitch,  hence  equal  to  '2  cos  A. 

Con'.MN  6.    Center  angle  for  Gear,  or  90°  —  A. 

Column  7.  Ten  times  the  angle  increment  for  Gear  of  10  teeth,  whiclr  of  course  equab  that  of  the 
engaging  Pinion. 

Column  8.    Diameter  increment  for  a  Gear  of  one  pitch,  hence  equal  to  2  sin  A. 

Use  of  Table.  —  In  columns  1  or  2  find  the  value  corresponding  to  the  ratio  of  given 
gears.     Against  this  value,  in  3  and  6,  the  center  angles  for  pinion  and  gear  are  given. 

The  angle  increment  may  be  found  by  dividing  the  value  in  4  by  the  number  of  teeth  in 
tiie  })inion,  or  l)y  dividing  the  value  in  7  by  the  number  of  teeth  in  the  gear. 

The  diameter  increment  for  the  pinion  is  obtained  by  dividing  the  value  in  .)  by  P,  and  that 
for  the  gear  by  dividing  the  value  in  8  by  P. 

The  value  of  the  angle  B  may  be  determined  with  sufficient  accuracy  by  making  it  |  of  B. 

*  The  plan  of  this  table  is  tliat  adopted  by  Mr.  George  B.  Grant.  See  "American  Machinist/'  Oct.  31,  1885,  and 
'•Odoiitics,"  page  itO. 


54 


BEVEL    GEAR    TABLE. 


BEVEL    GEAR    TABLE   FOR    SHAFTS    AT   90°. 


PROPORTION 
OF 

PINION. 

QEAS. 

A. 

B. 

2  E. 

900  — A. 

B. 

2  F. 

Angle 

Diameter 

Angle 

Diameter 

PINION 
TO  GEAR. 

(Center 
Angle. 

Increment. 
Divide 

Increment. 
Divide 

Center 
Angle. 

Increment. 
Divide 

Increment. 
Divide 

by  n. 

by  P. 

by  N.    ^ 

by  P. 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

1 

1 

1.000 

45. 

80.5 

1.414 

45. 

80.5 

1.414 

9 

10 

.900 

41.98 

76.0 

1.486 

48.02 

84.5 

1.337 

8 

9 

.888 

41.63 

75.6 

1.495 

48.37 

85.0 

1.329 

7 

8 

.875 

41.18 

75.0 

1.504 

48.82 

85.5 

1.317 

6 

7 

.857 

40.60 

74.1 

1.518 

49.40 

86.3 

1.302 

5 

6 

.833 

39.80 

73.0 

1.536 

50.20 

87.4 

1.280 

4 

5 

.800 

38.66 

71.1 

1.562 

51.34 

88.8 

1.249 

7 

9 

.777 

37.85 

70.0 

1.579 

52.15 

89.6 

1.228 

3 

4 

.750 

36.83 

68.5 

1.600 

53.17 

90.8 

1.200 

5 

< 

.714 

35.53 

66.2 

1.628 

54.47 

92.5 

1.162 

7 

10 

.700 

34.99 

65.1 

1.638 

55.01 

93.0 

1.147 

2 

3 

.666 

33.68 

63.2 

1.664 

56.32 

94.5 

1.109 

5 

8 

.625 

32.00 

60.4 

1.696 

58.00 

96.3 

1 .060 

\          3 

o 

.600 

30.96 

58.7 

1.715 

59.04 

97.3 

1.029 

^     1 

1    \'n 

7 

.571 

29.75 

o(}Aj 

1.736 

60.25 

98.5 

.992 

\U^''> 

9 

.555 

29.05 

55.4 

1.748 

60.95 

99.1 

.971 

— [il 

2 

.500 

26.56 

51.0 

1.789 

63.44 

101.4 

.894 

--^      4 

9 

.444 

23.94 

46.3 

1.827 

66.06 

103.6 

.812 

BEVEL  GEAR  TABLE. 


55 


BEVEL  GEAR  TABLE  FOR  SHAFTS  AT  90°. 


PROPORTION 

OF 

PINION 

TO  GEAR. 


1. 

2. 

3:    7 

.428 

2:    5 

.400 

3:    8 

.375 

1:    3 

.333 

3:10 

.300 

2:    7 

.285 

1:    4 

.250 

2:    9 

.222 

5 
11 

G 
13 

;    7 
;15 


.200 
.181 
.166 
.153 
.143 
.133 


1: 

8 

.125 

2 

17 

.117 

1 

9 

.111 

1 

10 

.100 

82.88  !  112.2 

83.30  1  112.3 

83.67  112.4 

84.30  1  112.6 


.248 
.233 
.221 
.200 


-^ 

n 
^  d' 

p-^. 

d' 
tan   A  =  ^, 

n 

tan  B 


2  sin  A 


2.25  sin  A 


tan  C  = 


2  E  =  p  cos  A  , 


2  F  =  p  sin  A. 


56 


BEVEL    GEARS    WITH    AXES    AT    ANY    ANGLE. 


63.  Bevel  Gears  with  Axes  at  any  Angle. 
If  the  axes  of  the  gears  intersect  at  angles  other 
than  90°,  the  drawing  of  the  bhmks  and  devel- 
o[)nient  of  the  teeth  do  not  differ  from  the  cases 
ah'eady  descriljed.  The  figuring  required  is  that 
inthcated  in  Fig.  27,  those  in  the  heavy  face 
being  used  to  determine  the  other  vahies,  and 
not  a[)pearing  on  the  finished  drawing. 


tan    A  = 


+  cos  a 


tan  A'  = 


—  +  cos  a 

N 


r,       2  sin  A           2  sin  A' 
tan   B  = or    


,       r.       2.25  sin  A          2.25  sin  A' 
tan  C  = or 


E  =  -  cos  A  ;      E'  =  —  cos  A'. 
P  P 


sin  A 


F'  =  —  sin  A'. 
P 


Fig.  27. 


Or,  the  values  for  E,  F,  E',  and  F  may  be  ob- 
tained from  the  table  for  shafts  at  90°,  pages  54 
and  55  by  determining  the  center  angles  A  and 
A',  and  finding  the  values  for  2  E  and  2  F,  corre- 
sponding to  each  gear  separately. 


WILLIS  S    ODOXTOGllAril.  5/ 

CHAPTER    VII. 

SPECIAL   FORMS    OF    ODONTOIDS,   NOTATION,   FORMULAS,   ETC. 

64.  Odontographs  and  Odontograph  Tables.  If  tooth  curves  are  to  be  drawn  according  to 
some  estal)lislied  system,  in  whieli  the  angle  of  pressure  is  constant,  or  but  one  diameter  of 
rolling  circle  be  used,  it  may  be  desirable  to  employ  some  of  the  approximate  methods  for 
shortening  the  operation.  While  it  is  unnecessary  for  the  student  to  familiarize  himself  with 
the  theor3%  or  even  the  details,  of  operating  the  various  systems  of  approximating  these  curves, 
it  is  essential  that  a  knowledge  be  had  of  the  more  useful  tables  and  methods  to  which  refer- 
ence may  be  made  when  required. 

Three  methods  are  employed  for  approximating  the  odontoidal  curves. 

First,  by  circular  arcs,  the  centers  and  radii  of  which  are  given  in  tables,  or  established  by 
instruments,  designed  for  this  purpose. 

Second,  by  curved  templets  from  \\liich  the  curves  may  be  traced  directly. 
Third,  by  ordinates. 

65.  Willis's  Odontograph.  Among  those  of  the  first  type,  the  oldest,  best  known,  and  least 
accurate,  are  tlie  odontographs  designed  by  Professor  Willis.  When  used  for  gears  having  a 
large  number  of  teeth,  the  error  is  very  slight ;  but  in  the  case  of  involute  teeth  of  small 
number  it  is  very  noticeable.      Fig.   28  illustrates  the  application  of  this  instrument  to  the 


58 


THE    GRANT    ODONTOCJRAPIIS. 


drawing  of  curves  of  the  cycloidal  system.  The'  centers  for  the  circuhir  arcs  designed  to 
approximate  the  curves  are  found  on  tlie  straight  edge,  A  B,  and  at  a  distance  from  the  zero 
point  of  the  scale  to  be  found  in  tlie  pul)lished  table  accompanying  the  instrument. 

'Jlie  theory  and  application  of  these  odontographs 
is  clearly  treated  of  in  the  instructions  accompanying 
these  instruments,  also  in  Stalil  and  Wood's  "  Elements 
of  Mechanism,"  pages  113  to  122,  and  more  briefly  in 
MacConUs  "  Kinematics,"  pages  172  to  174. 

66.    The  *'  Three  Point  Odontograph,"  designed  by 

Mr.  Geo.  B.  Grant,  is  a  table  for  face  and  flank  radii 

and  centers,  figured  for  circular  arcs  passing  through 

j  \        \        \  ^^^  three  most  important  points  of  the  tooth  curves ; 

/  \       \      !  viz.,   at   addendum    or   dedenduni  circles,   pitch   circle, 

and  a  point  midway  between.  This  gives  a  very  close 
approximation  to  the  true  curve  for  the  systeni  which 
has  radial  flanks  for  gears  of  twelve  teeth.  The  tables 
and  instructions  are  pul)lished  in  Grant's  "  Odontics," 
pages  41   and  42,  and  in  Stahl  and  Wood's  '•*  Elements  of  Mechanism,"  pages  124  and  125. 


Fig.  28. 


67.  The  Grant  Involute  Odontograph,  designed  by  Geo.  B.  Grant,  and  puljlished  in  his 
"Odontics,"  pages  29  and  80,  gives  a  very  close  approximation  to  the  involute  for  15°  angle 
of  pressure  and  epicycloidal  extension,  all  gears  being  designed  to  engage  a  12-tootlied  gear 
without  interference. 


THE    ROBIXSOX    AXD    KLP:iN    ODONTOCillAPHS 

68.  The  Robinson  Odontograph  differs  from 
the  preceding  in  that  it  is  an  instrument  hav- 
ing a  curved  edge  which  is  used  as  a  templet 
to  trace  the  tooth  curve,  tables  being  used  to 
determine  the  position  of  the  instrument  \^ith 
relation  to  the  pitch  circle. 

Fig.   29  illustrates    the    instrument  in  posi- 
tion.    The  curve   B  C  A  is  a  logarithmic  spiral, 
and  the  curve  B  F  H  the  evolute  of  the  first,  and 
therefore  a  similar  and  equal  spiral.     By  means 
of  this  instrument,  in  connection  with  the  pul>- 
lished  tables   accompanying   it,    involute   teeth  may  be  drawn 
as  well    as   C3^cloidal,  and    a   much   larger  range   of    the   latter 
is  possiljle  than    is  afforded  by  the  Willis  odontograi)h.      Tlu^ 
theory  of    this   instrument   is    best   treated    by  Professor   Rob- 
inson in  Van  Nostrand's  Eclectic  Magazine  for  July,  18T(>,  and 
Van   Nostrand's    "  Science    Series,"   No.   24.      Also   see  Stahl 
and  Wood's  "Elements  of  Mechanism,"  pages  12(3  to  1-30. 

69.  The  Klein  Coordinate  Odontograph.  Fig.  30  is  de- 
signed to  eliminate  the  labor  of  drawing  pitch  circles  of  large 
radii  by  constructing  the  cuinc  l)y  ordinates  from  a  radial 
line.       'I'he    tables    and    ex2)laiiatioii    of    tlie    method    may    l)e 


Fig.  30. 


60 


SPECIAL    FORMS    OF    ODONTOIDS    AND    THEIR    LINES    OF    ACTION. 


Fig.  33. 


found  in  Professor  Klein's  "  Elements  of  ^Machine 
Design,"  page  50. 

70.  Special  Forms  of  Odontoids  and  their  Lines 
of  Action.  Gears  maybe  classified  from  the  forms 
of  rack  teetli,  as  follows : 

System.  Tooth  Curve.  Lixe  of  Action-. 

Involute,      Fig.  31,  A  right  line,  .V  riglit  line. 

Cycloidal,    Fig.  o2,  A  cycloid,  A  circular  arc. 

Segmental,  Fig.  00.  A  circular  arc.  Conchoid  of  Xicomedes. 

In  like  manner  other  systems  might  be  derived 
from,  and  classified  by,  the  forms  of  their  rack 
teeth. 

It  is  of  interest  to  note  in  connection  with  the 
first  two  that  any  tooth  of  either  system  may  be 
derived  from  a  right  line.  In  the  cycloidal  system 
the  addendum  of  any  gear  tooth  will  properly 
engage  the  radial  flank  of  some  gear.  If.  there 
fore,  the  addenda  of  any  gear  tooth  be  made  to  fit 
the  dedenda  of  teeth  consisting  of  radial  flanks, 
the  resulting  teeth  must  be  cycdoidal.  \  skilled 
mechanic  with  file  and  straight-edge  could  in  this 


COXJUCIATE    CURVES. 

manner  })roduce  the  templet  for  any  de- 
sired cyeloidal  tooth  without  the  aid  of 
other  mechanism.  Of  course  such  a 
method  would  require  considerable  skill 
in  producing  a  perfect  tooth,  and  it  is  not 
the  best  means  to  the  end ;  but  it  is  of 
much  interest  to  the  student  as  illustrat- 
ing the  relation  between  the  mechanical 
and  graphic  methods  of  attaining  the  same 
end.  In  like  manner  we  may  produce  templets  for  invo- 
lute teeth  from  the  right  line  rack  tooth  of  the  system. 

71.  Conjugate  Curves. — The  curves  of  any  pair  of 
teeth  being  so  related  as  to  produce  a  uniform  velocity 
ratio  are  called  conjugate,  or  odontoids,  and  if  any  tooth 
curve  of  reasonal)le  form  be  assumed,  a  second  curve 
may  be  obtained  which  shall  be  conjugate  to  the  first. 
By  a  reasonable  form  is  meant  the  conformity  to  the 
following  principle :  — 

The  normals  to  the  curve  must  come  into  action 
consecutively,  as  in  Fig.  34,  and  not  as  in  Fig.  35,  in 
which  it  will  be  seen  that  the  normal  E  F  will  })ass 
through  the  pitch  point   M,  and  the  point  E  come  into 


62 


WORM    GEARING. 


Elg.  37. 


action  before  the  point  C,  which  is  impossible.  Let  C,  Fig.  36, 
be  any  tooth  form  conforming  to  the  above  condition,  and  the 
periphery  of  disk  A  its  pitch  line.  Suppose  it  is  required  to 
derive  its  conjugate  having  for  its  pitch  circle  the  periphery  of 
disk  B.  This  may  be  obtained  by  a  graphic  process,  as  in 
Art.  28,  page  15,  or  by  the  mechanical  method  known  as  the 
molding  process  of  Fig.  36.  C  is  a  templet  of  the  given  tooth 
form,  which  is  fastened  to  disk  A ,  and  revolving  in  contact  with 
disk  B,  the  disks  maintaining  a  constant  velocity  ratio.  The 
successive  positions  of  C  are  then  traced  on  the  plane  of  disk  B, 
and  the  tangent  curve  will  be  that  of  the  required  conjugate 
tooth. 

The  method  is  applicable  to  all  forms  of  spur  gear  teeth, 
but  to  only  one  form  of  bevel  gear,  the  octoid. 

72.  Worm  Gearing.  A  woi'm  is  a  screw  designed  to  oper- 
ate a  gear,  called  a  worm  wlieel  or  gear,  the  axis  of  the  latter 
being  perpendicular  to  that  of  the  worm.  Art.  3,  page  3. 
The  section  of  a  worm  and  gear  made  by  a  plane  perpendicular 
to  the  axis  of  the  gear,  and  including  the  axis  of  the  worm,  is 
identical  with  that  of  a  rack  and  gear  of  the  same  system  and 
pitch.  The  worm,  or  screw,  may  be  single,  double,  etc.  If 
single,  the   circular  pitch    corresponds  with   the   pitfli   of    the 


LITERATURE.  G3 

thread  :  if  double,  the  cireidai'  iiitcli  will  he  half  the  pitcli  of  the  thread,  etc.  To  avoid  niis- 
uiiderstandiug,  it  is  customary  to  speak  t)f  the  pitch  of  the  tluead  as  the  lead. 

A  drawing  of  the  tooth  foi-ni  is  required  oiiiy  in  special  cases  of  large  cast  gears,  and  the 
usual  representation  is  that  shown  by  Fig.  37. 

The  diameter  of  the  worm  is  connnonly  made  equal  to  foui'  or  live  times  the  circular  pitch, 
and  the  angle  A  varies  from  (iU  '  to  \H)  . 

FoiiMULAS    Full    WoilISI    AND    (iEAll. 

L    =  Lead  of  worm ; 
m  =  Threads  per  iucli  in  worm; 
d    =  Outside  diameter  of  worm; 
d'  "  Pitcli  diameter  of  worm; 
W  =  Wliole  diameter  of  gear; 
D   =--  Tliroat  diameter  of  gear; 
D'  =  Pitcli  diameter  of  gear; 

L    =  —  =  P',  for  single  tlireads, 

2 
L    =  —  =  2  P'.  for  doiihie  tlireads.  etc.; 
m 

Tj  and  r.,  are  dimensions  reciiiircd  for  the  liol),  or 

cutter,  emploved  in  cutting  the  worm  gear;  >.;        r^    ,    ^  /  ^ 

'        '     •'  "  •  W  =  D  +  2     rj  —  rj  cos  - 

C  =  Center  distance;  \  ^ 

73.  Literature.  The  following  list  of  books  and  articles  is  published  to  assist  the  student 
who  may  wish  to  puisue  the  subject  beyond  its  elementary  stage.  Only  those  treatises  have 
been  enumerated  which  are   likely  to  be  accessible  and   uscfid.      The  great  works  of  Willis, 


p,                  TT     U 

N  +  2' 

0..11-.^; 

D=^  +  ^ 

d       2 
•■i   -  2        P' 

^2  =  ^1  +  ^  P; 

r         D  +  d 
^   "        2 

1 
P' 

W  =  D  +  2  ( rj 

—  rj  cos 

(;4  LITEHATUHK.. 

Kaiikiiie,  and  Reuleux  are  omitted,  as  the  student  will  derive  more  beuetit  from  the  interpreta- 
tion of  these  works  by  hiter  authors  tlian  by  a  stndy  of  the  original  treatises. 

'^The  Mechanics  of  the  Machinery  of  Transmission,"  revised  by  Professor  Herrmann,  is 
Voh  Iir.,  Part  I.,  Sect.  1,  of  Weisbach's  "Mechanics  of  Engineering."  This  work  includes 
one  of  the  most  valualde  treatises  on  the  subject  of  gearing,  bnt  it  is  somewhat  dithcnlt. 
Wiley,  $')M. 

"■  Kinematics,"  by  Pi'ofessor  MacCord,  is  chiefly  devoted  to  the  snbject  of  gearing.  It  con- 
tains nuich  original  matter  of  importance.  No  student  of  the  subject  can  afford  to  do  witliout 
this  treatise.     Wiley,  -f!5.0(). 

"  Elements  of  Machine  Design,"  l)y  Professor  Klein,  was  published  for  the  students  of 
Lehigh  University.  Several  chapters  are  devoted  to  gearing,  and  include  some  excellent  tables 
and  problems.  The  Klein  coordinate  odontograph  is  fully  illustrated  and  explained.  J.  E. 
Klein,   Bethlehem,  Pa.,  -16.00. 

"  Odontics,"  by  Mr.  Geo.  B.  Grant,  is  one  of  the  most  valuable  modern  treatises  on  gearing. 
It  is  both  theoretical  and  practical.  It  is  concise,  contains  many  useful  tables,  and  is  well 
illustrated.  The  subject  cannot  be  })ursued  to  advantage  without  its  use.  Lexington  Gear 
Works,  Lexington,  Mass.,  <fl.OO. 

"  Practical  Treatise  on  Gearing,"  by  Mr.  O.  J.  Beale.  An  excellent  practical  treatment  of 
the  design  and  construction  of  gears.  It  deals  little  with  the  theoiy,  but  that  little  is  thor- 
oughly and  simply  taught.      Brown  &  Sliarpe  Manufactin-ing  Company,  Providence,  $L00. 

"  Eormidas  in  Geaiing."  This  is  published  by  the  Brown  &  Shari)e  Manufacturing  Com- 
])any,  and  contains  many  useful  formulas  for  the  draftsman,  and  valuable  hints  for  the  cutting 
of  frears.     .f2.00. 


NOTATION    AND    FORMULAS.  65 

"  Elementary  Alechainsin,'*  by  Professors  Stahl  and  Wood,  is  a  most  comprehensive  text 
book  on  the  subject  of  gearing.  It  is  well  classified,  contains  numerous  examples,  and  is  a 
valuable  reference  book  for  the  student.     Van  Nostrand,  '^2.00. 

The  student  is  recommended  to  read  the  following  articles  published  in  the  Ainei-ican 
Machinist. 

"Cutting-  Bevel  Gears  in  a  Universal  Milling  Machine."  by  O.  -I.  Beale,  June  20,  1895. 
"Planed  Bevel  Gear  Teeth,"  by  George  B.  Grant,  Dec.  9,  189G. 
"Grant's  Epicycloidal  Bevel  Gear  Generator,"  June  7,  1894. 
"  Bilgrani  Bevel  Gear  Cutting  Machine,"  May  9,  1885. 
"Bilgrani  Gear  Exhibit,"  Oct.  12,  1893. 

"  Bevel  Gear  Curves."  Chart  for  plotting  from  (iranfs  bevel  gear  chart,  by  H.  AValden,  Oct.  8  189(3 
(chart  corrected  Xov.  5,  1896). 

"The  Strength  of  Gear  Teeth,"  by  Henry  Hess,  Feb.  18,  1897. 
"Strength  of  Gear  Teeth,"  by  W.  T.  Sears,  June  10.  1897. 
"Gear  Arm  Proportions,"  by  Henry  Hess,  April  29,  1897. 

74.    Notation  and  Formulas. 

Spur  Gears. 

P'  =  Circular  pitch.  Art.  17,  page  11;  N  =  Number  of  teeth  in  gear; 

P   =  Diameter  pitch,  Ar.T.  18,  page  11;  n    =  Number  of  teeth  in  pinion; 

D'  =  Pitch  diameter  of  gear;  s    =  Addendum  of  tooth,  Aiix.  31,  page  17; 

D  =  Whole,  or  addendum,  diameter  of  gear;  f    =  Clearance,  Art.  27,  page  15; 

d'  =  Pitch  diameter  of  pinion:  t    =  Thickness,  Art.  31,  page  17; 

d   =  Whole,  or  addendum,  (liiimt'tei-  of  pinion;  p  ^  liCast  angle  of  pressure,  Aitr.  45,  page  8;?; 


G6 


NOTATIOX    AND    FORMrLAS. 


TT  -^  3.1416; 

N  D'       P' 


TT    D'  N  TT 

P   =   'TT-    •    ^,  =  ^, ,  Ai!T,  17,  page  11; 


N 
P   =  j^,,      P  P'  =  TT,  Akt.  18,  page  12; 


1 

P' 


f  =  Q  ~  o~B'  '■^''T-  31,  page  1/: 


P'   _      TT       _    1.57        . 

t    =  ^  ^  ^ ^ .  Art.  31,  page  1 . ; 

2        PD'+2       N4-2 
D=D'+2S  =  D'  +  ^  =  '^-y^-'  =  ^< 


_      /N  —  2 
'^°^  P  ~  V/  ~^'  Ai!T.  45,  page  33. 


Annular  Gears. 


Rg  ^  Radius  of  gear, 

Rp  =  Radius  of  pinion, 

fj     =  Inner  describing  circle, 

r.,    =  Outer  describing  circle, 

r^    =  Intermediate  describing  circle, 

C    =  Center  distance, 


'Art.  50,  page  ;19; 


Rg  =  r-  +  r, ,  Art.  50,  page  30: 

Rp  =  rg  —  r._, ,  Art.  50,  page  3!) 

C    =  Rg  —  Rp,  Art.  50,  page  39; 
fj  maximum  =  Rg  —  C  ;   rj  minimum  =  C,  Art.  52,  page  40; 
r.2  maximum  =  C  ;  r._>  minimum  =  2  C  —  Rg, 

Ai;t.  52,  page  40; 
r3  maximum  =  Rg  ;  r^  minimum  =  C,  Art.  51,  page  40. 


REV?:r.  Gears,  Shafts  at  90°,  Airr.  6i,  Page  5G. 

A  =  Tenter  angle  of  piiiiuu; 

B  =  Angle  increment; 

C  =  Angle  decrement; 

E  =  ()ne-half  the  (iianictci'  inciemenl  foi-  pinion: 

F         ( )n('-lialf  llic  di;nii('tcr  iiirrcinciit  fur  geai'. 


tan  A 

d'    _  n  . 
~  D'         N' 

tan  B 

2  sin  A. 

n 

tan  C 

2.25  sin  A 

n- 

E 

1           , 
=  p  cos  A 

sin  A. 


notation  and  formulas.  67 

Bevel  (iEaks,  Shafts  at  Othei:  than  *J0^,  Ai;t.  63,  Page  5(3. 

a    =  Aniile  of  .sliafl>;  ^       ,              sm  a 

°  tan  A    = 

A    =  Center  angle  of  pinion; 

A'  =  Cenlei  angle  of  gear;  ^^p  ^'  — 

B    =  Angle  inerennMil : 

„          .       ,     ,  _,        2  sin  A        2  sin  A 

C    =  Angle  deeienienl :  tan  B  = 


N 

— 

+ 

cos 

a 

11 

sin  a 

n 

N 

+ 

COS 

a 

n  N 

E    =  One-half  the  diameter  inerenient  for  pinion;  ^        22.5  sin  A  2  25  sm  A' 

tan  C   = =- , 

n  N 

E'  =  OiH'-half  the  diameter  increment  lor  gear;  ■,  , 

E  =  _  cos  A  E'  =  p  cos  A'; 

F    =  Dimension  i-e<|nireil  for  hacking  of  jiinion; 

c        74-  •  •      1  .■      1       1  ■         i-  r   =  „  sm  A  F    =  —  sm  A  ; 

I-    =  Dimension  re(iuued  tor  hacking  ot  gear.  P  P 

\Voi:m  (iEAEs,  Ajrr.  72,   I'AciE  (32. 

L    =  Lead  of  worm ;  ,  1         r^    .  ,      ,         , 

L    =  —  =^  P'  tor  .sinule  threads; 
m 

nn  =  Threads  per  inch  in  worm;  2 

L    =  —  =  2  P'  tor  doiihle  thread,  etc., 
m 
d    =  Outside  diameter  of  worm;  k,  m     i    9 

D'=  -  •  D  =       "*"     : 

P  P 

d    =  Pitch  diameter  of  worm: 

TT    D  ..,„,/  A' 


P'  -- •        W  =  D  +  2     1^  -  ri  COS 

D   =  Thread  diameter  of  gear;  N   +  2  y 

C  =  ^-±-  -  -• 
D'  =  Pitch  diameter  of  gear;  2  P' 

W  =  AVhole  diameter  of  gear.  ^^   ~  7        P        '        "^i    ~  ''i  """  3  '^^ 


68  METHOD  TO  BE  OBSERVED  IX  PEKEOKMJXC;  THE  PKOBLEMS. 

/ 

CHAPTER    VIII. 

PROBLEMS. 

75.  Method  to  be  Observed  in  Performing  the  Problems.  No  attempt  slionld  be  made  to 
oraphically  solve  the  following  problems  until  the  general  principles  involved  are  well  under- 
stood. 

The  first  requisite  to  tliis  is  tlie  mastery  of  C-hapter  II.,  on  Odontoidal  Curves;  and  this 
can  be  best  acquired  l)y  the  drawing  of  the  various  curves,  together  A\'ith  a  study  of  their 
characteristics.  No  problems  have  been  given  on  this  topic,  but  the  following  course  of 
study  would  be  desirable :  — 

Having  prescribed  diameters  for  rolling  circles  and  director,  or  pitch  circles,  draw  a  cycloid, 
epicycloid,  and  hypocycloid,  as  described  in  Arts.  5,  G,  and  7,  page  5.  Obtain  a  sufficient 
number  of  points  in  each  case  to  enable  the  curves  to  l)e  drawn  free-hand  with  considerable 
accuracy,  after  wliich  they  may  be  corrected  by  the  use  of  scrolls.  Next  prescribe  a  point  on 
each  (not  one  already  found),  and  draw^  normals  to  each  by  Akt.  8,  page  5. 

The  second  method,  Art.  9,  page  5,  is  the  more  practical,  and  shonld  also  l)e  studied  by 
drawing  a  small  part  of  each  curve,  beginning  at  a  point  on  the  director  circle. 

It  is  also  desirable  that  one  of  the  epitrochoidal  forms  be  drawn,  and  a  normal  determined. 
Art.  11,  page  6. 

The  problems  are  designed  to  be  solved  on  a  sheet  which  shall  measure  10"  by  14"  within 


I 


PKOBLKM     1.       CYf'LOIDAL    LIMITING    CASE.  69 

the  margin  line,  and  the  hiy-out  of  these  sheets  is  given  on  Plates  14  and  15,  there  being 
four  problems  on  each  plate.     ]\Ieasurements  are  from  the  margin  line. 

It  is  unnecessary  to  represent  all  the  teeth  in  a  gear,  but  such  as  are  shown  should  be 
drawn  with  the  greatest  accuracy  attainable  by  the  student.  Without  this  care  the  study  will 
avail  one  little,  and  the  time  consumed  in  discovering  errors  will  be  great. 

The  inking  of  the  curves  may  be  omitted  if  time  will  not  admit  of  its  being  w^ell  done  ; 
but  in  either  case  it  is  desirable  to  emphasize  the  curves,  and  distinguish  clearly  between  the 
gears  by  making  a  very  light  wash  of  color  on  the  inside  of  the  curve,  the  width  to  be  about 
one-quarter  of  an  inch.     One  color  may  be  used  for  the  pinion,  and  a  second  for  the  rack  and 

Cycloidal  Limiting  Case.       Face  or  Flank  only. 

d' 


gear. 

Problem  i,  Plate 

14, 

Fig. 

Example. 

D' 

1 

10 

2 

10 

3 

121 

4 

5 

10 

6 

lOL 

7 
8 

12' 

N 

n 

A 

B 

15 

12 

H 

41 

*2 

12 

H 

4' 

21 

3 

4 

15 

10 

H 

4i 

15 

H 

4i 

14 

10 

3] 

H 

24 

12 

2'» 

•^4 

4 

21 

12 

31 

4i 

Statement  of  Problem.  Having  given  the  diameters  of  pitch  circles,  number  of  teeth, 
and  diameter  of  describing  circle,  it  is  re(]^uired  to  draw  the  teeth  for  pinion,  gear,  and  rack, 
liaving  arcs  of  contact  equal  to  the  pitch,  and  contact  on  one  side  of  pitch  point  only. 


70  PROBLEM     1.       CYCLOIDAL    LIMITIXC;    CASE. 

Study  Arts.  1  to  26  before  performing  this  prol)lein. 

Operations.  1.  By  Art.  18,  page  11,  determine  the  value  of  N,  n,  D'  or  d,  one  of  whicli  is 
omitted  from  the  table.      Observe  that  -  =  -• 

d  n 

2.  Draw  center  and  pitch  lines  and  describing  circle.  Lay  off  the  circular  })itch  on  each 
gear  by  spacing  the  circumferences  into  as  many  parts  as  there  are  teeth. 

3.  Obtain  the  first  point  of  contact  by  laying  off  from  the  pitch  point  on  the  describing 
circle  an  arc  equal  to  the  circular  pitch,  the  directif)n  being  determined  b}^  the  rotation 
required.     Art.  16,  page  10.     Ar.t.  21,  page  12.     Arts.  22  and  23,  page  13. 

1.  With  the  above  describing  point,  generate  the  face  and  flank  required.  Arts.  14  and 
15,  page  10. 

5.  Draw  the  working  faces  of  gear  teeth,  and  assuming  the  gear  teeth  to  l)e  pointed,  draw 
opposite  side  of  each.     Art.  16,  page  10. 

6.  Draw  the  working  flanks  of  the  pinion  teeth,  observing  that  the  depth  must  be  sufificient 
to  admit  the  gear  teeth,  but  without  clearance.  Obtain  the  thickness,  and  draw  the  opposite 
sides.     Art.  16,  page  10. 

7.  Draw  the  describing  circle  for  rack.  Obtain  the  first  point  of  contact  between  pinion 
and  rack,  and  describe  the  cycloid  for  rack  teeth.  Construct  rack  teeth.  Ar/r.  2o,  i)age  14. 
Note  that  thickness  of  rack  tooth  must  equal  space  between  pinion  teeth,  or  thickness  of 
gear  teeth,  measured  on  the  })itcli  line. 

8.  To  determine  points  of  contact  of  conjugate  teeth,  assume  any  ])oint  on  face  of  gear 
tooth,  and  determine,  first,  its  position  when  in  contact  with  the  pinion ;  second,  the  jjoint 
of  the  pinion  tooth  engaging  it.  Since  the  contact  must  take  place  on  the  path  of  contact. 
Art.  21,  page  12,  the  assumed  })oint  will  lie  at  the  intersection  of  this  arc  and  one  described 


PROBLEM    2.       CYCLOIDAL    LIMITING    CASE.  71 

through  the  given  point  from  center  of  gear.  To  solve  the  second,  describe  an  arc  from  the 
center  of  the  pinion  through  the  point  previously  determined,  and  its  intersection  with  the 
pinion  flank  will  be  the  engaging  point  required. 

Next  construct  the  normals  for  each  of  these  points.  Art.  8,  page  5.  They  should  he 
equal  to  each  other,  and  also  to  the  distance  from  the  pitch  point  to  the  point  on  the  path  of 
contact  in  which  they  engage.     Art.  14,  page  10. 

9.  Obtain  the  maximum  angle  of  obliquity,  or  pressure,  between  gear  and  pinion,  pinion 
and  rack.     Art.  24,  page  14. 

Problem  2,  Plate  14,  Fig.  2.  Cycloidal  Limiting  Case.  Face  and  Flank.  Study  Arts. 
2(3  to  30. 

Statement  of  Problem.  The  diameters  of  gears,  number  of  teeth,  and  describing  circles 
being  given,  it  is  required  to  draw  the  teeth  for  pinion,  gear,  and  rack,  when  the  arc  of 
approach  =  the  arc  of  recess  =  half  the  circular  pitch,  the  flank  of  gear  being  radial. 

Operations.  1.  Draw  center  lines,  pitch  lines,  and  rolling  circles,  the  second  circle 
being  determined  by  Art.  9,  page  0.  Divide  the  pitch  circle  into  the  required  parts  to  obtain 
the  circular  pitch. 

2.  Lay  off  arcs  equal  to  —  on  each  of  the  rolling  circles  to  obtain  the  first  and  last  points 
of  contact,  observing  the  direction  of  rotation  prescriljed  in   Fig.  2. 

3.  With  the  point  thus  determined  on  small  rolling  circle,  describe  the  addendum  of  gear 
tooth  and  dedendum  of  pinion  tooth.  With  the  point  on  the  second  describing  circle  generate 
the  addendum  of  pinion  tootli.  The  dedendum  of  gear  tooth  being  radial  may  then  be  drawn. 
Make  the  dedenda  of  pinion  and  gear  deep  enough  to  admit  the  engaging  addenda,  but  allow 
no  clearance. 


72  PROBLEM  .?.   CYCLOIDAL  GEAR. 

4.  Draw  the  working  faces  of  the  pinion  teeth  and  then  the  opposite  faces  to  make  the 
teeth  pointed.  Simihirly  draw  the  gear  teeth,  making  them  pointed  also.  The  sum  of  the 
thickness  of  tlie  teeth  cannot  he  greater  that  the  circular  pitch.  Art.  29,  page  16.  In  this 
case  it  will  he  found  to  be  about  one-hundredth  of  an  inch  less,  which  will  be  the  backlash. 
An  increase  in  the  diameter  of  either  rolling  circle  would  make  the  solution  impossible. 

5.  Draw  the  dedenda  of  pinion  and  gear  teeth. 

6.  The  describing  circles  for  the  rack  teeth  Avill  be  determined  by  Art.  14,  page  10. 
Draw  the  circles  with  their  centers  on  the  line  of  centers,  and  obtain  the  first  and  last  points 
of  contact.  These  points  should  fall  on  the  addendum  and  dedendum  of  pinion  teeth  already 
drawn,  as  in  Plate  5  at  M  and  0  .  From  these  points  describe  the  addenda  and  dendenda  of 
the  rack  teeth.     The  thickness  of  these  teeth  must  equal  those  of  tlie  gear. 

7.  Obtain  the  maximum  angle  of  pressure  for  approach  and  recess  between  pinion  and 
gear  and  pinion  and  rack.  It  would  also  be  desirable  to  obtain  the  curve  of  least  clearance 
in  one  case.     Art.  28,  page  15. 

Problem  3,  Plate  14,  Fig.  3.  Cycloidal  Gear.  Practical  Case.  ('om})lete  Chapter  III.  be- 
fore performing  this  problem. 

N          n  A          a         C 

15  12  4        5       31 

21  12  4  4  3| 
20  16  3i  4  3 

22  12  3i  5  4 

16  12  4"  5  3i 
20  12  3  4  4^ 


^^X  AMPLE 

d' 

1 

9 

2 

8 

3 

10 

4 

8 

5 

9 

6 

7 

PROBLEM    3.        CYC'LOIDAL    GEAR.  73 

Statement  of  Problem.  Tlu'  diameters  of  pitch  circles  and  rolling  circles  being  given, 
and  the  number  of  teeth  known,  it  is  required  to  draw  the  teeth  for  gear,  pinion,  and  rack, 
to  obtain  the  maximum  angle  of  obliquity,  and  the  arcs  of  approach  and  recess  in  each  case. 
The  teeth  will  be  standard  with  j^^"  backlash.     Art.  31,  page  17.     Art.  71,  page  61. 

Operations.  1.  Figure  the  diameter  of  gear,  circular,  and  diametral  pitch.  Arts.  17 
and  18,  page  11,  and  determine  proportions  of  teeth.     Art.  31,  page  17. 

2.  Draw  center  lines,  pitch  lines,  addendum,  and  dedendum  circles,  and  rolling  circles. 
Divide  the  pitch  circle  into  as  many  parts  as  there  are  teeth,  beginning  to  space  at  the  pitch 
point. 

3.  Beginning  at  the  pitch  point,  describe  pinion  flank,  gear  face,  gear  flank,  and  pinion 
face,  by  Art.  9,  page  5.     See  also  Art.  34,  page  21. 

4.  Lay  off  thickness  of  teeth.  Art.  31,  page  17,  and  describe  addenda  of  pinion  and  gear 
teeth  by  approximate  method.  Art.  34,  page  22.  Describe  dedenda  by  Art.  10,  page  11. 
Draw  fillets.     Art.  31,  page  18. 

5.  Describe  rack  teeth. 

6.  Determine  the  following  for  gear,  pinion,  and  I'ack  in  tei-ms  of  P'.  Arts.  21  to  24 
inclusive,  pages  12,  13,  and  14,  Art.  32,  page  18. 

Pinion  and  Gear.                      Pinion  and   Rack. 
Arc  of  approach    

Arc  of  recess 

Arc  of  contact  

Maxinunn  angle  of  pressure  ^ 


/4  PROBLEM    4.       INVOLUTE    LIMITING    CASE. 

Problem  4,  Plate   14,  Fig.  4.     Involute  Limiting  Case.     Study  Arts.  38  to  42. 

Statement  of  Problem.  Number  of  teeth  five  and  six.  Pinion  teetli  pointed.  No 
Ijacklasli  or  clearance.     Arc  of  contact  equal  to  the  circular  pitch. 

This  problem  being  similar  to  that  of  Plate  8,  reference  will  be  made  to  that  figure. 

The  case  being  a  limiting  one,  the  distance  between  the  points  of  tangency  of  base  circles 
and  line  of  pressure  must  equal  one-sixth  of  the  circumference  of  the  gear  base  circle,  or  one- 
tifth  of  the  circumference  of  the  pinion  base  circle.      The  tangent  of  the  angle  of  pressure 

will  equal  ^  =  ^  =     /  ^       ,  but  A  D  =  D  K  C  by  construction,  and   D  K  G  =  tt.     Also  A  F  + 

D  C  =  5i,  hence,  —= — —  =  —  =  tan.  of  the  angle  of    pressure.     The  angle  corresponding  to 

this  tangent  is  29°  44'    6".     The   distance   between   the   centers  will   be  ^ho'^  +  A  F  +  'D~G^  = 

The  angle  of  pressure  and  distance  between  centers  could  have  been  determined  graphi- 
cally by  laying  off  F  A ,  in  any  direction,  equal  to  the  radius  of  pinion  base  circle,  A  D  perpen- 
dicular to  FA,  and  equal  to  one-fifth  of  pinion  base  circle.  Finally,  D  G  perpendicular  to  A  D, 
and  equal  to  the  radius  of  gear  l)ase  circle. 

Operations.  1.  Draw  the  line  of  centers,  base  circles,  and  line  of  pressure.  Deter- 
mine the  points  of  tangency,  which  limit  the  action  in  either  direction,  and  through  the  pitch 
point,  determined  by  the  intersection  of  the  line  of  centers  and  line  of  pressure,  draw  the 
pitch  circles.  It  is  desirable  now  to  test  A  D  by  proving  it  equal  to  one-fifth  of  the  pinion  base 
circle,  or  one-sixth  of  the  gear  l)ase  circle. 

2.  Draw  the  involute  A  C,  Plate  8,  of  the  gear,  and  D  p  of  the  pinion.  Airr.  12,  page  7. 
Art.  38,  page  26.  Determine  the  circular  pitch,  and  lay  off  as  many  divisions  as  there  are 
teeth  to  be  drawn.      Copy  the  curves  already  drawn. 


PROBLEM  .-,.   INVOLUTE  PRACTICAL  CASE.  75 

3.  Draw  the  opposite  face  of  pinion  tectli.  making  them  pointed.  To  draw  the  opposite 
faces  of  gear  teeth  proceed  as  follows :  Since  contact  between  the  opposite  faces  must  take 
place  along  the  line  of  action  C  E ,  Plate  8,  the  contact  between  the  engaging  teeth  will  be 
at  E.  At  E  draw  arc  E  1  from  center  G  .  Bisect  this  arc,  and  lay  off  M  and  H  from  this  radial 
bisector  equidistant  with  A  and  C.  'I'hrough  these  points  describe  the  curve  of  opposite  face, 
and  draw  the  remaining  teeth. 

That  j)ortion  of  the  teeth  lying  within  tlie  base  circle  will  ])e  radial,  and  extend  sufficiently 
to  admit  the  engaging  teeth,  but  without  clearance. 

4.  Construct  two  rack  teeth.     Art.  40,  page  29. 

5.  Epicycloidally  extend  the  gear  teeth  so  as  to  make  them  pointed.  Similarly  extend  the 
rack  teeth,  l)ut  only  as  mucli  as  the  clearance  for  tlie  pointed  gear  tooth  will  permit.  Art. 
41,  page  30. 

Problem  5,  Plate  15,  Fig.  i.     Involute  Practical  Cases.     Complete  the  study  of  Chapter  IV. 

State.mext  of  Pnop.LEMS.  Several  gears  and  racks  are  given  to  describe  involute  teeth 
of  standard  dimensions.  'J'o  determine  the  interference,  if  there  be  any,  and  to  correct  the 
curves  for  the  same. 

QPERATroxs.  1.  Draw  three  or  four  teeth  of  gear  A,  and  two  teeth  of  engaging  pinion 
B,  the  angle  of  pressure  being  16°.  Art.  42,  page  32,  Fig.  16.  Make  contact  at  pitch 
point  in  all  cases.  Correct  for  interference  by  epicycloidal  extension.  Art.  31,  page  17. 
Art.  42,  page  30.     Art.  41,  page  32. 

2.  Draw  three  or  four  teeth  of  o-ear  A  enofaofino-  rack  F . 

3.  Draw  three  teeth  of  pinion  B  engaging  rack  E,  and  correct  rack  teeth  for  interference. 


76 


PROBLEM    6.       CYCLOIDAL    ANNULAR    GEAR. 


4.  Draw  a  portion  of  gear  C  and  rack  K ,  the  angle  of  pressure  being  20°.     Test  this  for 
interference  by  Art.  45,  page  33,  as  well  as  by  graphic  method. 

5.  Draw  a  few  teeth  of  gear  D,  the  angle  of  pressure  being  15°.     Determine  the  least 
number  of  teeth  that  will  engage  it  without  interference. 

Problem  6,  Plate  15,  Fig.  2.     Cycloidal  Annular  Gear.     Study  Arts.  48  to  56. 


EXAMPLK. 

D' 

d' 

N 

n 

A 

a 

B 

1 

m 

9 

13 

6 

7 

H 

H 

2 

191 

9 

13 

6 

6h 

4 

51 

3 

191 

9 

13 

6 

6 

41 

51 

4 

171 

7 

15 

6 

7 

31 

51 

5 

17i 

7 

15 

6 

7i 

3 

5h 

Statement  of  Problem.  The  number  of  teeth  and  diameters  of  pitch  and  describing 
circles  being  given,  it  is  required  to  draw  the  tooth  outlines,  and  determine  the  increased  arc 
of  contact  due  to  secondary  action.  The  arc  of  contact,  not  including  that  due  to  the 
secondary  action,  is  equal  to  the  circular  pitch,  and  the  arc  of  approach  equals  the  arc  of 
recess. 

Operations.     1.    Draw  the  center  and  pitch  lines  and  describing  circles. 

2.  Determine  the  circular  pitch,  and  lay  off  half  this  amount  from  the  pitch  point  on  each 
of  the  describing  circles  to  determine  the  first  and  last  points  of  contact. 

3.  Describe  the  curves  of  the  teeth. 


PROBLEM    7.       INVOLUTE    ANNULAR    GEAR.  i  I 

4.  Determine  the  intermediate  describing  cnrve,  and  draw  tlie  same  to  obtain  the  limit  of 
secondary  action. 

5.  Determine  the  maximnni  angle  of  pressure  for  approach  and  recess.     Also  the  angle 
of  pressure  for  the  last  point  of  secondary  action,  and  the  increase  in  the  arc  of  contact. 

Problem  7,  Plate  15,  Fig.  2.     Involute  Annular  Gear.     Complete  Chapter  V. 


Example. 

D' 

d' 

N 

n 

Angle 

of  Pressure. 

B 

1 

15 

-i 

20 

10 

20° 

61 

2 

15 

G 

30 

12 

15° 

7 

3 

16 

8 

16 

8 

20° 

6 

4 

20 

8 

30 

12 

15° 

7 

5 

24 

18 

24 

18 

20° 

oh 

Stateiment  of  Problem.  The  pitch  diameters,  number  of  teeth,  and  angle  of  pressure 
being  given,  it  is  required  to  draw  the  tooth  cui've,  to  determine  if  there  Avill  l)e  any  inter- 
ference when  the  addenda  of  pinion  teeth  are  made  standard,  and  finally  the  length  of  the  arc 
of  contact  in  terms  of  P'. 

Operations.     1.    Draw  center  and  pitch  lines,  line  of  pressure,  and  base  circles. 

2.  Make  addenda  of  pinion  standard  if  a  second  engagement  does  not  take  place.  Art. 
56,  page  43,  and  limit  addenda  of  gear  by  Art.  56,  page  43. 

3.  Determine  the  arc  of  contact  in  terms  of   P'. 


78 


PROBLEM    8.       CYCLOIDAL    AND    INVOLUTE    BEVEL    GEARS. 


Problem  8,  Plate  15,  Fig.  3.     Cycloidal  and  Involute  Bevel  Gears.     Shafts  at  90^.     Study 
Arts.  57  to  63. 


IMPLE. 

,  P 

N 

n 

Q 

K 

1 

3 

18 

15 

31 

u 

2 

4 

24 

20 

31 

n 

3 

2 

16 

12 

H 

n 

4 

4 

28 

20 

3 

n 

5 

3 

21 

15 

3 

u 

6 

2 

14 

12 

n 

^ 

7 

3 

21 

18 

•^8 

u 

8 

2 

18 

14 

4 

n 

9 

4 

20 

16 

H 

1 

10 

3 

21 

18 

3| 

u 

w 


3 
4 

u 

3 

3 
4 

n 

21 

1 

2 

4' 

1 

91 

"2 

i 

li 

2^ 

"^8 

3 
4 

11 

31 

1 

1^ 

3i 

3 
4 

2 

4 

f 

H 

91 

~4 

1 

li 

3i 

1 

li 

3 
4 

n 

n 

1^ 

1 

H 

1 

n 

1 

u 

7 

8 

n- 

1 

la 

3 

4 

1 

I 

1* 

I 


If  involute,  make  angle  of  pressure  15°. 

If  cycloidal,  make  diameter  of  rolling  circles  equal  to  the  elements  of  normal  cone  of 
pinion. 

Statement  of  Problem.  The  proportions  of  the  gear  being  given  by  the  table,  it  is 
required  to  draw  the  gear  blanks,  describe  the  development  of  the  teetli  on  the  normal  cones, 
and  figure  the  gears. 

Operations,  1.  Having  determined  the  pitch  diameters,  draw  the  gear  blanks.  Art. 
60,  page  48. 


PROBLEM    9.       CYCLOIDAL    AM)    INVOLUTE    BEVEL    (iEARS.  79 

2.  Describe  two  or  three  teeth  of  each  gear  on  the  developed  surfaces  of  the  outer  and 
inner  normal  cones.      Akt.  60,  page  48. 

3.  Figure  the  gears,  Art.  61,  i)age  51. 

Problem  9,  Plate  15,  Fig.  4.     Cycloidal  and  Involute  Bevel  Gears.     Shafts  at  other  than 
90°.     Study  Art.  63. 


EXAMT 

'Lt; 

a 

P 

N 

n 

Q 

J 

K 

L 

M 

H 

w 

1 

40° 

b 

24 

15 

9 

8^ 

2h 

i 

li 

i 

u 

2 

45° 

3 

24 

15 

9 

"s 

2i 

i 

u 

5 

n 

3 

50° 

4 

34 

24 

m 

S 

2 

i 

n 

i 

n 

4 

55° 

3 

27 

21 

9 

8 

21 

^ 

li 

i 

IS 

5 

60° 

2 

20 

12 

8t. 

li 

2n 

5 

s 

If 

1 

2 

V 

Y 

2.1 

u 

2 

u 

1^1 
1* 

1] 

Is 

1§ 

3^.       i 

If  involute,  make  angle  of  pressure  15°. 

If  cycloidal,  make  diameter  of  rolling  circles  equal  to  the  elements  of  normal  cone  of 
pinion. 

Statement  of  Problem.  Tlie  pro|)ortions  of  the  gear  being  given  by  the  table,  it  is 
required  to  draw  the  gear  ])lanks,  describe  the  teeth  on  the  development  of  the  normal  cones, 
and  figure  the  gear. 

Operations.  1.  Determine  the  ])it{h  diameters  from  al)ove  tal)le,  and  draw  tiie  gear 
blanks. 

2.  Describe  two  or  three  teeth  of  each  gear  on  tlie  developed  surfaces  of  the  outer  and 
inner  normal  cones. 

o.    Figure  the  geai-s. 


I  ]^  D  E  X„ 


Addendum  defined,  12:  proportion  for,  17. 

Angle  decrement,  52. 

Angle  increment,  52. 

Angle  of  edge,  51 ;  of  face,  51. 

Angle  of  obliquity,  or  pressure,  14 ;  affected  by  rolling  circle, 
18;  constant,  28;  for  involute,  31;  influence  of,  33; 
method  for  determining,  33;  reduced  in  annular  gear- 
ing, 40. 

Annular  gear,  notation,  and  formulas,  ()(>:  epicycloidal  prol)- 
lem,  76;  involute  problem,  77. 

Annular  gearing,  38;  secondary  action  in.  ."W ;  interchan- 
geable with  spur  gearing,  42;  involute  system  of.  43. 

Approaching  action  detrimental,  17. 

Approximate  cycloidal  curves.  22. 

Approximation,  Tredgold,  47  ;  by  circular  arcs,  22 

Arc  of  approach  defined,  13. 

Arc  of  contact  defined,  13;  relation  to  circular  pitch,  ](>. 

Arc  of  recess  defined,  13. 

Backing,  .52. 

Back  cone,  48. 

Backlash  defined,  Hi;  dimen.sions  for,  18. 

Base  circle  defined,  7,  27. 

Base  of  system,  21 ;  in  annular  gearing,  42. 

Beale's  "Practical  Treatise  on  Gearing,"  04. 


Bevel  gear  defined,  2:  Theory  of,  45;  character  of  curves 
employed,  4(5;  drafting  the,  48;  blank,  49;  length  of 
face,  49;  figuring  the,  51 ;  table  for,  53,  54,  55;  cliart  for 
plotting  curves,  65 ;  notation  and  formulas,  66 :  prol> 
lems,  78,  79. 

Bevel  gears  with  axes  at  any  angle,  5(5. 

Bilgram,  Hiigo,  inventor  of  octoid  tooth,  47;  machine  for 
cutting  bevel  gear  teeth,  47,  65;  exhibit,  65 

Brown  &  Sharpe  publications,  (54. 

Circular  pitch  defined,  11. 

Character  of  curves  in  bevel  gearing,  46. 

Clearance  defined,  15;  proportion  for,  17. 

Clock  gears,  17. 

Conchoid  of  Nicomedes,  (50. 

Conditions  governing  the  practical  case,  16. 

Conjugate  curves  defined,  9,  61. 

Constant  angle  of  pressure,  28. 

Constant  velocity  ratio  defined,  1. 

Conventional  representation  of  spur  gears,  25. 

Contact,  point  of,  5;  radius,  5;  path  of,  12;  arc  of,  13. 

Coordinate  odontograph,  .59. 

Crown  gear,  47. 

Curtate  epitrochoid,  (i. 

Curve  of  least  clearance,  15. 


81 


82 


INDEX. 


Curves,  odontoidal,  4. 
Cutting  bevel  gear  teetli,  65. 
Cutting  angle,  51. 

Cycloid,  defined,  4;  problem  relating  to,  68. 
Cycloidal  action,  Theory  of,  8. 

Cycloidal  curves,  second  method  for  describing,  5 ;  approxi- 
mated, 22. 
Cycloidal  system  of  annular  gearing,  38. 
Cycloidal  annular  gear  problem,  76. 
Cycloidal  bevel  gear  i^roblem,  78,  7fl. 
Cycloidal  limiting  case  problems,  (59.  71. 
Cycloidal  practical  case  problem,  72. 

Dedendum  defined,  12;  proportions  for,  17. 

Defects  of  involute  system,  35. 

Describing  circle,  defined,  4;  a  path  of  contact,  12;  maxi- 
mum and  minimum,  16;  influence  on  sliape  and  effi- 
ciency of  teeth,  18 ;  relation  to  interchangeable  gears,  20. 

Describing  disk,  8. 

Describing  point,  4. 

Describing  cone,  45. 

Describing  cylinder,  45. 

Describing  radius,  5. 

Description  of  bevel  gear  table,  53. 

Developed  pitch  circle,  50. 

Development  of  normal  cone,  49. 

Diameter  pitch,  11. 

Director  circle,  5. 

Double  contact  in  annular  gearing,  3fl. 

Double  generation  of  epicycloid  and  hypocycloid,  6. 

Drafting  bevel  gears,  48. 

"  Elements  of  Machine  Design,"  (i4. 


"  Elementary  Mechanism,"  65. 

Epicycloid,  defined,  5;  second  method  for  describing,  5; 
double  generation,  6;  spherical,  45;  problem  relating  to, 
68. 

Epicycloidal  extension,  30. 

Epitrochoid,  defined,  6;  curtate,  6;  prolate,  7:  problem  re- 
lating to,  68. 

Exterior  (outer)  describing  circle,  39;  limitations  of,  40,  41. 

Face  gearing,  2. 

Face  of  gear,  24. 

Face  of  tooth,  12. 

Flank  of  tooth,  12;  radial,  18. 

Figuring  bevel  gears,  51. 

Fillet,  18  ;  size  of,  18. 

"  Formulas  in  Gearing,"  64. 

Formulas  for  worm  and  gear,  63. 

Formulas,  Notation  and,  65. 

Gearing,  1. 

Gear  arm  proportions,  65. 

Gears,  interchangeable,  20:  face  of,  24;  comparison  of,  24. 

Generating  point,  4. 

Generating  radius,  5. 

Grant,  Geo.  B.,  bevel  gear  chart,  53;  three  point  odonto- 
graph,  58;  involute  odontograph,  58;  "  Odontics,"  58, 
64;  epicycloidal  and  bevel  gear  generator,  65. 

Hyperboloid  of  revolution,  2. 

Hyperbolic  gears,  2. 

Hypocycloid,  defined,  5;  second  method  for  describing,  5;  a 
radial  line,  6:  double  generation,  (!;  si)lierical,  45;  prob- 
lem relating  to,  68. 


INDEX, 


83 


Influence  of  the  angle  of  pressure,  33. 

Influence  of  the  diameter  of  rolliiij;  circle  on  shai>e  and 
efficiency  of  teeth,  18. 

Inner  desci'ibino;  circle,  3i);  limitations  of,  40,  41. 

Inner  normal  cone,  48,  50. 

Instantaneous  radius,  4. 

Intermediate  describing  circle,  39;  limitations  of,  40. 

Internal  gear,  see  annular  gear. 

Interference,  32;  in  annular  gearing,  43. 

Interchangeable  gears,  20. 

Involute,  4 ;  defined,  7 ;  system,  26 ;  curves,  character  of,  27 ; 
rack,  28 ;  system  of  annular  gearing,  43 ;  annular  gear 
l)roblem,  77;  bevel  gear  tooth,  46;  bevel  gear  problems, 
78,  79 ;  limiting  case,  29 ;  limiting  case  jiroblem,  74 ;  prac- 
tical case,  30;  practical  case  problem,  75. 

Involute  action,  Theory  of,  26;  limit  of,  28. 

Involute  gearing,  defects  of  system,  35. 

Involute  teeth,  epicycloidal  extension  of,  30. 


"  Kinematics,"  MacCord's,  (i4. 
Klein's  coordinate  odontograph,  59; 
Design,"  64. 


Elements  of  Machine 


Law  of  tooth  contact,  10. 

Lead  of  screw,  04. 

Least  angle  of  i)ressure,  method  for  determining,  33. 

Least  number  of  teeth  in  annular  gears,  42. 

Limit  of  involute  action,  28. 

Limiting  ca.se,  cycloidal,  10.  14;  involute,  29;  ainiular  gear- 
ing, 38. 

Limitations  of  intermediate,  exterior,  and  interior  describing 
circle,  40,  41. 

Line  of  action  a  great  circle,  47. 


Literature,  63. 
Logarithmic  spiral,  59. 

ISIacCord's  "Kinematics,"  64. 

Method  for  determining  least  angle  of  pressure,  33. 

Method  to  be  observed  in  performing  problems,  68. 

"Mechanics  of  Engineering,"  64. 

"  Mechanics  of  the  Machinery  of  Transmission,"  64. 


Normal,  defined,  4  ;  to  construct,  5  :  law 
Normal  cone,  48 ;  development  of,  49. 
Notation  and  formulas,  65. 


governing,  Gl. 


Obliquity,  angle  of,  14. 

Octoid  bevel  tooth,  47,  62. 

"  Odontics,"  Grant's,  64. 

Odontoid  defined,  1 ;  special  forms  of,  60. 

Odontoidal  curves,  4 ;  problems  relating  to,  (i8. 

Odontographs  and  odontograph  tables,  57. 

Odontograph,  "Willis,  57:  Grant  involute,  58;  Grant  Three 

point,  58;  Robinson,  59;  Klein,  59;  coordinate,  59. 
Outer  describing  circle,  39;  limitations  of,  40. 
Outer  normal  cone,  48. 

Path  of  contact  defined,  12;  affected  by  rolling  circle,  18;  a 

right  line,  28. 
Path  of  approach  defined,  15. 
Path  of  recess  defined,  15. 
Pitch  cone,  48. 
Pitch  line,  10. 
Pitch  point,  9,  10,  27. 
Pitch,  circular,  11 :  diameter,  11. 
Planed  bevel  gear  teeth,  ()5. 


84 


INDEX. 


Positive  rotation  defined,  11. 

Practical  case,  conditions  governing  the,  16;  cycloidal,  21 

involute,  30;  annular,  42. 
"Practical  Treatise  on  Gearing,"  04. 
Pressure,  angle  of,  14. 
Prolate  epitrochoid,  7. 
Proportions  for  standard  tooth,  17. 
Problems,  method  to  be  observed  in  performing,  68. 

Rack,  14 ;  involute,  28 ;  gears  classified  by,  60. 

Radial  flank,  18 ;  as  base  of  system,  21,  60. 

Radius,  describing,  5 ;  contact,  5. 

Rankine,  64. 

Reuleux,  64. 

Robinson  odontograph,  59. 

Rolling  circle,  see  describing  circle. 

Rotation,  jiositive,  1. 

Screw  gearing  defined,  3. 

Scroll,  use  of,  11. 

Second  method  for  describing  cycloidal  curves,  5. 

Secondary  action  in  annular  gearing,  38,  41. 

Segmental  system,  60. 

Skew  gear  defined,  3. 

Spiral  gear  defined,  3. 

Special  forms  of  odontoids,  45. 


Spherical  epicycloid,  45. 

Spherical  hypocycloid,  45. 

Spur  gear  defined,  2;  illustrated,  10;  having  action  on  one 
side  of  pitch  point,  10;  having  action  on  both  sides  of 
pitch  point,  14;  conventional  representation,  25 ;  inter- 
changeable with  annular  gears,  42 ;  notation  and  formu- 
las, 05. 

Theory  of  cycloidal  action,  8. 
Theory  of  involute  action.  20. 
Thickness  of  tooth,  17. 
Three  point  odontograpli,  58. 
Tooth  contact,  law  of,  10. 
To  construct  a  normal,  5. 
Tredgold  approximation,  47. 

Unsymmetrical  teeth,  37. 
Use  of  bevel  gear  table,  53. 

Velocity  ratio  constant,  1 ;  not  aeffcted  by  increase  of  center 
distance  in  involute,  28. 

Weisbacli's  "Mechanics,"  64. 

Willis,  odontograph  of,  57  ;  writings  of,  64. 

Worm  gearing  defined,  3,  62;  notation  and  formulas  for,  67. 

Worm  wheel,  62. 


I 


Plate  I. 

Cycloid,  Epicycloid,  Hypocycloid  and  Involute  curves. 

ItEFEIlENCES    TO    TEXT, 

Art.  4,  Page  4.  Art.  8,  Pao-e  5. 

5,        ^     4.  9,  5. 

G,             5.  12,  7. 
7,             5. 


Plate  1 

, 

B 

1 

i__^ 

B 

/ 

"^ 

5v 

kl?" 

v\ 

"7^ 

^C^^l^^x 

j 

/  V^ 

\\L^ 

/\ 

^0. 

\                 1 

/     \ 

/  2V^  \      k^^^' 

N. 

\ \^  5 

'    \ 

i     >^ 

i-         'xL 

A 

g~ — \- 

-  Kr 

'f- / 

---At 

/ 

^^-\ — ~~~- 

1 

|l\v^^ 

r  / 

"  \ 

/        1 

\  /^^   \ 

^^ 

\ 

'  \ 

yc 

Y 

J 

"/        J 

\               \7c  I  1/       A^,^ 

vV 

\ 

\ 

\, 

Ax 

^ 

1  ^ 

V!>< 

J^V^/ 

3/v  K 

1^ 

^  'e 

c 

D' 

B'" 

Je 

/V 

^'0 

FIG 

1 

j 

/        ^~~^^7<^:i 

-^^^Z""i 

^y^-y^ 

/O^ 

^\         /B,ic^ 

0/ 

1             / 

./ 

\ 

E 

C           1               /'                /— 

""^/V^^^^           /           /Fig 

2 /^rVr 

-^/A^\ 

1        >^ 

A" 

^ 

\ 

/ 

1  /^    \ 

/     A 

/       \ 

1 

^^^"--^ 

\ 

[ 

f^~~Jl 

n  \    a""^^'^^'"'          / 

/ 

A' 

^^ 

-^A. 

ri 

i\ 

I^nM-^ 

/ 
/ 

A 

{B 

/ 

/ 

.     N 

v^ 

^^ 

^^,^#^A^/V]\^-4?^  p/ 

/ 
/ 

k 

/ 

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/         \ 
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4 

■///  K 

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K 

Fig. 3 

\0 

Y 

p 

P 

P 

Plate  2. 

Epitrochoidal  curves.     Double  generation  of  Epicycloid  and 
Hypocycloid.     Approximate  method. 

REFERENCES    TO    TEXT. 

Art.  10,   Page     6. 
11,  6. 

34,  22. 


Plate   2 


Plate  3. 

Mechanical  method  for  describing  Odontoidal  curves. 

REFERENCES    TO    TEXT. 

Art.  13,  Page     8. 
15,  10. 

21,  12. 


Plate  3, 


Plate  4. 

Cycloidal  Gear,  Pinion  and  Rack  having  action  on  one  side  of 
pitch  point.     Limiting  case. 


Airr, 


REFERENCES    TO    TEXT. 


IT).    Page   10. 

Art. 

21, 

Page  11. 

10,        '      10. 

25," 

11. 

18,               12. 

26, 

14. 

19,             12. 

36, 

24. 

20,             12. 

47, 

37. 

23,             13. 

LATE    4. 


Plate  5. 

Cycloidal  Gear,  Pinion  and  Rack  having  action  on  both  sides 
of  the  pitch  point.     Limiting  case. 

ItEFEEENCES    TO    TEXT. 

Art.  23,  Pao-e  1-3.  Art.     3(3,  Page  24. 

20,  '      14.  41),  38. 

28,  15.  PK015.     2,  72. 

32,  10. 


Plate   5. 


Plate  6. 

Cycloidal  Gear,  Pinion  and  Rack.     Practical  case. 

REFERENCES    TO    TEXT. 

Art.  34,  Page  21. 

36,  24. 

37,  25. 


Plate   6. 


Plate  7. 

Involute  Gear  and  Pinion.     Limiting  case.     Mechanical  method 
for  describing  the  Involute. 

REFEREXCES    TO    TEXT. 

Art.  38,  Page  26. 
39,  27. 

42,  31. 


Plate  7. 


Plate  8. 

Involute  Gear,  Pinion  and  Rack.     Limiting  case. 

REFERENCES   TO    TEXT. 

Art.  39,  Page  28.  Art.     41,  Pao-e  30. 

40,  29.  Pror.     4,  74. 


PLATE    8. 


Plate  9. 

One  Pitch  Involute  Gear  and  Pinion,  showing  Interference. 

REFERENCES    TO    TEXT. 

Art.  42,  Page  30.  Art.  44,  Page  33. 

43,  32.  46,  35. 


Plate  9. 


1   PITCH  INVOLUTE  GEAR  &  PINION 
SHOWING  INTERFERENCE 


Plate  10, 

One  Pitch  Involute  Pinion  and  Rack,  showing  Interference. 

REFERENCES    TO    TEXT. 

Art.  42,    Page  30.  Art.  44,  Page  33, 

43,  32.  46,  35. 


Plate  to. 


RACK 


1   PITCH  INVOLUTE  PINION  &  RACK 
SHOWING  INTERFERENCE 


Plate  II. 

Annular  Gearing. 

REFERENCES    TO    TEXT. 

Art.  49,  Page  38.  Ai;t.  52,  Page  41. 

50,  39.  54,  42. 


Plate  11 


Plate  12. 

Annular  Gearing.     Special  cases. 

REFERENCES    TO    TEXT. 

Art.  50,  Page  39. 

51,  40. 

52,  40. 


Plate  12 


0  12  3 

116        1  I  .  I 

ililinlilililililililililihlililililililililihlililihiilil' 


Plate  13. 

Bevel  Gearing. 

REFERENCES    TO    TEXT. 

Art.  50,  Page  47.  Art.  60,  Page  48. 


Plate  13 


Plate  14. 

Problems  i  to  4  inclusive. 

EEFEREXCES    TO    TEXT. 

Art.     65,  Page  69.  Peob.  3,  Page  72. 

Peob.     1,  69.  4,  74. 

2,  71. 


Plate  14 


I  \  /<^        DRIVER 


T 


A 


t  / 


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\ 


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__B ^  ^/         \ 

-9/  \ 


v/ 

9 


.Fig.  1 


^i/ 


RADIAL   FLANKS 
ON    GEAR. 


\ 


\. 


■Fig.  2' 


// 


/ 


/.--~\..  //•' 


> 


>  I    \  /        A       / 

/  I  /        >-a   nJ; 

/  ^/      \^    -■ -- 


-6.334 


/ 


'^^■^ch'linS. 


qt/- 


PITCH    LINE   OF   RACK 


._.         Y^ 


\- 


/  r\ 


/ 


/ 


//y       \    /    NX/ft 

-H-^-^: — # -^ 

^V     /     "-^  /7V\        /I 


X< 


^v/ 


/ 


PITCH    UNtlOF^RACK 


Fig.  3 


/ 


Fig.  4 


Plate  15. 

Problems  5  to  9  inclusive. 

REFERENCES    TO    TEXT. 

Art.     65,  Page  69.  Prop..  7,  Page  77. 

Prob.     5,  75.  8,  78. 

6,  76.  9,  79. 


Plate  15 


o^^^^?-^e!^.o,. 


PlTCH'LiNE    OF 


University  of  California 

SOUTHERN  REGIONAL  LIBRARY  FACILITY 

Return  this  material  to  the  library 

from  which  it  was  borrowed. 


B     000  003  239 


•  ^'A'f^^  ^  »>  tQii^iii 


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Univers 
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